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It is shown that the rings under the title admit an order-theoretical characterization as in the commutative case studied by Abian. Introduction. Let R be an associative ring equipped with the binary relation (^) defined by xay if and only if xy = x2 in R. In this paper, it is shown that ( ^ ) is an order relation on R if and only if, R has no nilpotent elements i9*0). Conditions on the binary relation (g) in order that R split into a direct product of division rings are then studied in the light of Abian's result (l, Theorem l). Using similar argumentation and using certain subdirect representation of rings with no nilpotent elements, one obtains a complete similarity with the commutative case (yet, no extra complication in the computa- tions). Conventions. R is an associative ring which is, unless otherwise stated, with no nilpotent elements (other than 0). As a result of (2), R can be embedded into a direct product of skewdomains, R—* YLiei £i (that is to say, rings R, having no one-sided divisors of zero). The former embedding is fixed throughout the paper. It is therefore legiti- mate to identify any element x in R with the family consisting of all its projections (xj.e/. Finally, all definitions in (l) are extended (verbatim) to the present case (of a noncommutative ring R) and are freely used throughout.

A ring R is called, according to [2], a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring.In this short note we shall give some additional remarks to A. Hattori [2]. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in [2], which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.

In the above-named paper (this JOURNAL, vol. 77 (1955), pp. 895-913), Theorem 6. 4 should have been stated only for rings of zero characteristic: the argument for the case of prime characteristic breaks down in the last formula line on p. 911. This involves jettisoning Theorem 6. 5 (the implied "proof " of which depended on applying Theorem 6. 4 to homomorphs which cannot be guaranteed to have zero characteristic, even when the given ring has). However, the writer has no evidence that Theorem 6. 4 as stated or Theorem 6. 5 is actually false. And in any case, since the valid part of the proof of Theorem 6. 4 establishes the weaker conclusion [Xm, ynp?] = 0 without characteristic hypothesis, the remarks following Theorem 6. 4 still hold good, provided that we modify the final parenthetical clause so as to read "which would at any rate imply that every weak K-ring R with kc, m, n satisfying (a) or (b) has R/J commutative."

In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of noncommutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The noncommutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition, however, defines a larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on the examples and extend our studies beyond counting. We enrich our invariants with the following structures: the inclusion of subcategories makes them partially ordered sets and considering semi-orthogonal pairs of subcategories as edges amounts to directed graphs. It turns out that the problem for counting $D^b(A_k)$ in $D^b(A_n)$ has a geometric combinatorial parallel - counting of maps between polygons. Estimating the numbers counting noncommutative curves in $D^b({\mathbb P}^2)$ modulo the group of autoequivalences, we prove finiteness and that the exact determining of these numbers leads to a solution of Markov problem. Via homological mirror symmetry, this gives a new approach to this problem. Regarding the structure of a partially ordered set mentioned above, we initiate intersection theory of noncommutative curves focusing on the case of noncommutative genus zero. The above-mentioned structure of a directed graph (and related simplicial complex) is a categorical analogue of the classical curve complex, introduced by Harvey and Harrer. The paper contains pictures of the graphs in many examples and also presents an approach to Markov conjecture via counting of subgraphs in a graph associated with $D^b({{\mathbb{P}}}^2)$. Some of the results proved here were announced in a previous work.

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a complete answer is given in the case where R is local: it is proved in this paper that every ideal of R is isomorphic to a trace ideal if and only if R is an artinian Gorenstein ring, or a 1-dimensional hypersurface with multiplicity at most 2, or a unique factorization domain.

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the $${H^\infty}$$ functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form $$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$ where $${e_{\ell}, {\ell} = 1, 2, 3}$$ are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables $${x = (x_{1}, x_{2}, x_{3})}$$ and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version $${T^{\alpha}, {\rm for} \alpha \in (0, 1)}$$ , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.

This paper is a very brief and gentle introduction to non-commutative geometry geared primarily towards physicists and geometers. It starts with a brief historical description of the motivation for non-commutative geometry and then goes on to motivate the subject from the point of view of the the understanding of local symmetries affordee by the theory of groupoids. The paper ends with a very rapid survey of recent developments and applications such as non-commutative toric geometry, the standard model for particle physics and the study of the Riemann Hypothesis., Comment: 21 pages, 2 figures, To appear in the Notices of the American Mathematical Society

In this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves. Examples are provided by the algebras of global sections of quantizations of symplectic resolutions, quantum Hamiltonian reductions, and spherical symplectic reflection algebras. We introduce the notion of holonomic modules for such algebras. We show that, provided the algebraic fundamental groups of all leaves are finite, the generalized Bernstein inequality holds for the simple modules and turns into equality for holonomic simples. Under the same finiteness assumption, we prove that the associated variety of a simple holonomic module is equi-dimensional. We also prove that, if the regular bimodule has finite length, then any holonomic module has finite length. This allows one to reduce the Bernstein inequality for arbitrary modules to simple ones. We prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions, for quantum Hamiltonian reductions and for Rational Cherednik algebras. The paper contains a joint appendix by the author and Etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution.

Let $R$ be a commutative Noetherian ring and $\alpha$ an automorphism of $R$. This paper addresses the question: when does the skew polynomial ring $S = R[\theta; \alpha]$ satisfy the property $(\diamond)$, that for every simple $S$-module $V$ the injective hull $E_S(V)$ of $V$ has all its finitely generated submodules Artinian. The question is largely reduced to the special case where $S$ is primitive, for which necessary and sufficient conditions are found, which however do not between them cover all possibilities. Nevertheless a complete characterisation is found when $R$ is an affine algebra over a field $k$ and $\alpha$ is a $k$-algebra automorphism - in this case $(\diamond)$ holds if and only if all simple $S$-modules are finite dimensional over $k$. This leads to a discussion, involving close study of some families of examples, of when this latter condition holds for affine $k$-algebras $S = R[\theta;\alpha]$. The paper ends with a number of open questions., Comment: 23 pages; comments welcome

In this paper, Normalized Weighted Bonferroni Mean (NWBM) and Normalized Weighted Bonferroni Harmonic Mean (NWBHM) aggregation operators are proposed. Besides, we check the properties thereof, which include idempotency, monotonicity, commutativity, and boundedness. As the intuitionistic fuzzy numbers are used as a basis for the decision making to effectively handle the real-life uncertainty, we extend the NWBM and NWBHM operators into the intuitionistic fuzzy environment. By further modifying the NWBHM, we propose additional aggregation operators, namely the Intuitionistic Fuzzy Normalized Weighted Bonferroni Harmonic Mean (IFNWBHM) and the Intuitionistic Fuzzy Ordered Normalized Weighted Bonferroni Harmonic Mean (IFNONWBHM). The paper winds up with an empirical example of multi-attribute group decision making (MAGDM) based on triangular intuitionistic fuzzy numbers. To serve this end, we apply the IFNWBHM aggregation operator.

In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of Lichnerowicz, Gromov and Lawson, and Connes on the non-existence of metrics of positive scalar curvature. In this paper we review in more detail the material needed for the proof of our theorem and we extend our non-existence results to non-compact manifolds of bounded geometry. We also give a first obstruction result for the existence of metrics with (not necessarily uniform) leafwise PSC in terms of the A-hat class in Haefliger cohomology.

Hochschild (co)homology and Pirashvili's higher order Hochschild (co)homology are useful tools for a variety of applications including deformations of algebras. When working with higher order Hochschild (co)homology, we can consider the (co)homology of any commutative algebra with symmetric coefficient bimodules, however traditional Hochschild (co)homology is able to be computed for any associative algebra with not necessarily symmetric coefficient bimodules. In a previous paper, the author generalized higher order Hochschild cohomology for multimodule coefficients (which need not be symmetric). In the current paper, we continue to generalize higher order Hochschild (co)homology to work with associative algebras which need not be commutative and in particular, show that simplicial sets admit such a generalization if and only if they are one dimensional., revision includes one dimensional result

In the present paper, we expand the domain of work on theconcept of semiderivations in 3-prime near-rings through the study ofstructure and commutativity of near-rings admitting semiderivations sat-isfying certain differential identities. Moreover, several examples havebeen provided at places which show that the assumptions in the hypothe-ses of various theorems are not altogether superfluous. 1. IntroductionThroughout this paper, N is a zero-symmetric left near ring. A near ringN is called zero symmetric if 0x= 0 for all x∈ N (recall that in a left nearring x0 = 0 for all x∈ N). N is called 3-prime if xNy = {0} implies x= 0or y = 0. The symbol Z(N) will represent the multiplicative center of N,that is, Z(N) = {x∈ N | xy= yxfor all y∈ N}.For any x,y∈ N; as usual[x,y] = xy−yxand x◦y= xy+yxwill denote the well-known Lie product andJordan product, respectively. Recall that N is called 2-torsion free if 2x= 0implies x= 0 for all x∈ N. For terminologies concerning near-rings we referto G. Pilz [7].An additive mapping d: N → N is said to be a derivation if d(xy) = xd(y)+d(x)yforall x,y∈ N, orequivalently, asnotedin [8], that d(xy) = d(x)y+xd(y)for all x,y ∈ N. An additive mapping d: N → N is called semiderivation ifthere exists a function g : N → N such that d(xy) = xd(y) + d(x)g(y) =g(x)d(y)+d(x)yand d(g(x)) = g(d(x)) for all x,y∈ N.Obviously, any deriva-tion is a semiderivation, but the converse is not true in general (see [6]). Therehas been a greatdeal of workconcerning derivations in near-rings(see [1, 2, 4, 5]where further references can be found). In this paper, we study the commuta-tivity of addition and multiplication of near-rings. Two well-known results forderivations in near-rings have been generalized for semiderivation. In fact, ourresults generalize some theorems obtained by the authors together with Rajiin [1].

In this paper we demonstrate thatnon-commutative localizationsof arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category$\text{}\underline{C}$by a set$S$of morphisms in the heart$\text{}\underline{Hw}$of a weight structure$w$on it one obtains a triangulated category endowed with a weight structure$w^{\prime }$. The heart of$w^{\prime }$is a certain version of the Karoubi envelope of the non-commutative localization$\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$(of$\text{}\underline{Hw}$by$S$). The functor$\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of$S$invertible. For any additive category$\text{}\underline{A}$, taking$\text{}\underline{C}=K^{b}(\text{}\underline{A})$we obtain a very efficient tool for computing$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$coincides with the ‘abstract’ localization$\text{}\underline{A}[S^{-1}]$(as constructed by Gabriel and Zisman) if$S$contains all identity morphisms of$\text{}\underline{A}$and is closed with respect to direct sums. We apply our results to certain categories of birational motives$DM_{gm}^{o}(U)$(generalizing those defined by Kahn and Sujatha). We define$DM_{gm}^{o}(U)$for an arbitrary$U$as a certain localization of$K^{b}(Cor(U))$and obtain a weight structure for it. When$U$is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general$U$the result is completely new. The existence of the correspondingadjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over$U$.

In Chajda's paper (2014), to an arbitrary BCI-algebra the author assigned an ordered structure with one binary operation which possesses certain antitone mappings. In the present paper, we show that a similar construction can be done also for pseudo-BCI-algebras, but the resulting structure should have two binary operations and a set of couples of antitone mappings which are in a certain sense mutually inverse. The motivation for this approach is the well-known fact that every commutative BCK-algebra is in fact a join-semilattice and we try to obtain a similar result also for the non-commutative case and for pseudo-BCI-algebras which generalize BCK-algebras, see e.g. Imai and Iseki (1966) and Iseki (1966).

In the present paper, necessary conditions for the metric and topological projectivity of closed ideals of Banach algebras are given. In the case of commutative Banach algebras, a criterion for the metric and topological projectivity of ideals admitting a bounded approximate identity is obtained. The main result of the paper is as follows: a closed ideal of an arbitrary C*-algebra is metrically or topologically projective if and only if it admits a self-adjoint right identity.

In a recent paper, we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper, we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm–Liouville hypergroups, etc. A non-commutative hypergroup is also considered.

Let {\mathcal B} be a linear space of matrices over a field {\mathbb spanned by n\times n matrices B_1, \dots, B_m. The non-commutative rank of {\mathcal B}$ is the minimum r\in {\mathbb N} such that there exists U\leq {\mathbb F}^n satisfying \dim(U)-\dim( {\mathcal B} (U))\geq n-r, where {\mathcal B}(U):={\mathrm span}(\cup_{i\in[m]} B_i(U)). Computing the non-commutative rank generalizes some well-known problems including the bipartite graph maximum matching problem and the linear matroid intersection problem. In this paper we give a deterministic polynomial-time algorithm to compute the non-commutative rank over any field {\mathbb F}. Prior to our work, such an algorithm was only known over the rational number field {\mathbb Q}, a result due to Garg et al, [GGOW]. Our algorithm is constructive and produces a witness certifying the non-commutative rank, a feature that is missing in the algorithm from [GGOW]. Our result is built on techniques which we developed in a previous paper [IQS1], with a new reduction procedure that helps to keep the blow-up parameter small. There are two ways to realize this reduction. The first involves constructivizing a key result of Derksen and Makam [DM2] which they developed in order to prove that the null cone of matrix semi-invariants is cut out by generators whose degree is polynomial in the size of the matrices involved. We also give a second, simpler method to achieve this. This gives another proof of the polynomial upper bound on the degree of the generators cutting out the null cone of matrix semi-invariants. Both the invariant-theoretic result and the algorithmic result rely crucially on the regularity lemma proved in [IQS1]. In this paper we improve on the constructive version of the regularity lemma from [IQS1] by removing a technical coprime condition that was assumed there.

This paper is devoted to presenting a MacWilliams type iden-tity for m-spotty RT weight enumerators of byte error control codes overfinite commutative Frobenius rings, which can be used to determine theerror-detecting and error-correcting capabilities of a code. This providesthe relation between the m-spotty RT weight enumerator of the code andthat of the dual code. We conclude the paper by giving three illustrationsof the results. 1. IntroductionThe error control codes play an important role in improving reliability incommunications and computer memory system [5]. Recently, high-densityRAM chips with wide I/O data, called a byte, have been increasedly usedin computer memory systems. These chips are very likely to have multiplerandom bit errors when exposed to strong electromagnetic waves, radio-activeparticles or high-energy cosmic rays. To make these memory systems morereliable, spotty [21] and m-spotty [20] byte errors are introduced, which can beeffectively detected or corrected using byte error-control codes. To make clearthe error-detecting and error-correcting capabilities of a code, the research hasbeen done on some special types of polynomials, called weight enumerators.In general, the weightenumeratorofacode is apolynomialdescribingcertainproperties of the code, and an identity which relates the weight enumerator of acode with that of its dual code is called the MacWilliams type identity. For thepast few years, various weight enumerators with respect to m-spotty HammingWeight (Lee weight and RT weight) have been studied for various types ofcodes. Suzuki et al. [19] proved a MacWilliams type identity for binary byteerror-controlcodes. M. Ozenand V. Siap[8] andI. Siap[17] extended this result¨

It is well known that the Gaussian symplectic ensemble is defined on the space of \(n\times n\) quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices based on the exact known form of the density function of the eigenvalues (see Erdős in Russ Math Surv 66(3):507–626, 2011; Erdős in Probab Theory Relat Fields 154(1–2):341–407, 2012; Erdős et al. in Adv Math 229(3):1435–1515, 2012; Knowles and Yin in Probab Theory Relat Fields, 155(3–4):543–582, 2013; Tao and Vu in Acta Math 206(1):127–204, 2011; Tao and Vu in Electron J Probab 16(77):2104–2121, 2011). Due to the fact that multiplication of quaternions is not commutative, few works about large-dimensional quaternion self-dual Hermitian matrices are seen without normality assumptions. As in natural, we shall get more universal results by removing the Gaussian condition. For the first step, in this paper, we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to the semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.

If for every elements x and y of an associative algebraic struc-ture (S,·) there exists a positive integer r such that ab = b r a, then S iscalled quasi-commutative. Evidently, every abelian group or commuta-tive semigroup is quasi-commutative. Also every finite Hamiltonian groupthat may be considered as a semigroup, is quasi-commutative however,there are quasi-commutative semigroups which are non-group and noncommutative. In this paper, we provide three finitely presented non-commutative semigroups which are quasi-commutative. These are thefirst given concrete examples of finite semigroups of this type. 1. IntroductionThe quasi-commutativity property in algebraic structures is one of the inter-esting ideas which has been studied by many authors since 1971. The classifi-cation or identification of certain major classes of semigroups has been studiedas well. For more and detailed descriptions on the quasi-commutative semi-groups, quasi-commutative Hamiltonian semigroups, quasi-commutative superHamiltonian semigroups and periodic Hamiltonian semigroups one may con-sult the prolific articles [3, 6, 8, 9, 10, 11]. For our purposes, we need torecall the notion of presentation hX |Ri of formal generators X and relatorsR where, hX |Ri is defined properly for finitely generated semigroups or forfinitely generated monoids. Note that, R is called the set of relations whenhX |Ri is a group presentation. Furthermore, some preliminaries and moreinformation on the semigroups and monoids presentation are required in thepresentation theory of semigroups which may be found in [4, 5, 7]. The naturalquestion which may be posed here is how the non-group quasi-commutativesemigroups may be constructed? In this paper, we construct three finitelypresented non-commutative semigroups which are not groups and show thatthey are finite and quasi-commutative. This construction is based on exam-ining possible semigroup presentations of the known quasi-commutative finite

We derive in the paper the tensor product functor −⊗ R − byusing proper GP C -resolutions, where C is a semidualizing module. Aftergiving several cases in which different relative homologies agree, we use thePontryagin duals of G C -projective modules to establish a balance resultfor such relative homology over a Cohen-Macaulay ring with a dualizingmodule D. 1. IntroductionUnless otherwise stated, throughout this paper R is a commutative ringwith identity, and all modules are unitary. We denote by P the class of pro-jective modules. For a module M, the Pontryagin dual or character moduleHom Z (M,Q/Z) is denoted by M ∗ .In 2009, Sather-Wagstaff et al. [16] introduced and studied several relativecohomology functors with respect to a fixed semidualizing module C (see Sec-tion 2 for definition), such as Ext iGP C M (−,−) derived from Hom R (−,−) usingproper GP C -resolutions of the first variable and Ext iMGI C (−,−) derived fromHom R (−,−) using proper GI C -coresolutionsof the second variable, where GP

Let $(R,\frak m)$ be a commutative noetherian local ring. In this paper, we prove that if $\frak m$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $\frak m$ is a direct summand of (a direct sum of) syzygies of $M$. Applying this result to the case where $\frak m$ is quasi-decomposable, we obtain several classfications of subcategories, including a complete classification of the thick subcategories of the singularity category of $R$., Comment: 18 pages. Some minor changes throughout the paper; statement of Corollary 6.5 improved; Remark 6.7, Corollary 6.8, and reference [2] are added. Final version to appear in the Mathematical Proceedings of the Cambridge Philosophical Society

The purpose of this paper is to introduce the concept of strongly extending modules which are particular subclass of the class of extending modules, and study some basic properties of this new class of modules. A module M is called strongly extending if each submodule of M is essential in a fully invariant direct summand of M. In this paper we examine the behavior of the class of strongly extending modules with respect to the preservation of this property in direct summands and direct sums and give some proper- ties of these modules, for instance, strongly summand intersection property and weakly co-Hopflan property. Also such modules are characterized over commutative Dedekind domains.

In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares in the polynomial ring in the topology induced by the ℓ1-norm is equal to Pos([–1; 1]n), the cone consisting of all polynomials that are non-negative on the hypercube [–1,1]n. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted -seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theoremfrom2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of 2d-powers, proving, in particular, that for any integer d ≥ 1, the closure of the cone of sums of 2d-powers in the topology induced by the -norm is equal to Pos([–1; 1]n).

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod-Shafarevich algebras. This paper provides bounds for the growth function on images of Golod-Shafarevich algebras based upon the number of defining relations. This extends results from [32], [33]. Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky [7] by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov [40]. Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss in [8]., Comment: arXiv admin note: text overlap with arXiv:1207.6503

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say Γ(M), such thatwhen M = R, Γ(M) is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F.Anderson and S. B. Mulay, in [6], have been generalized for Γ(M) in thepresent article. We show that Γ(M) is connected with diam(Γ(M)) ≤ 3.We also show that for a reduced module M with Z(M) ∗ 6= M \ {0},gr(Γ(M)) = ∞ if and only if Γ(M) is a star graph. Furthermore, weshow that for a finitely generated semisimple R-module M such that itshomogeneous components are simple, x,y∈ M\ {0} are adjacent if andonly if xRTyR = (0). Among other things, it is also observed thatΓ(M) = ∅ if and only if M is uniform, ann(M) is a radical ideal, andZ(M) ∗ 6= M\{0}, if and only if ann(M) is prime and Z(M) ∗ 6= M\{0}. 1. IntroductionAll rings in this paper are commutative with identity and all modules areunitary right modules. Let G be an undirected graph. We say that G isconnected if there is a path between any two distinct vertices. For distinctvertices x and y in G, the distance between x and y, denoted by d(x,y), is thelength of a shortest path connecting x and y (d(x,x) = 0 and d(x,y) = ∞ ifno such path exists). The diameter of G isdiam(G) = sup{d(x,y) | x and y are vertices of G}.A cycle of length n in G is a path of the form x

In this paper, G denotes a metacyclic 2-group of positive type of nilpotency class at least three and Ω is the set of all subsets of commuting elements of G of size two in the form of (a, b), where a and b commute and lcm(|a|, |b|) = 2. The probability that a group element of G fixes a set is one of the generalizations of the commutativity degree that has been recently introduced. In this paper, the probability that an element of fixes a set for metacyclic 2-groups of positive type of nilpotency class at least three is computed. The results obtained are then applied to graph theory, more precisely to the orbit graph and generalized conjugacy class graph.

The authors used the notion of Nil clean rings, which was introduced by P. V. Danchev and W. Wm. McGovern in their paper “Danchev P.V., McGovern W.Wm., Commutative weakly nil clean unital rings, J. Algebra 425 (2015), 410–422.”. But unfortunately, it was not cited in the paper. So the reference no. 2 of the paper should be replaced by the paper of P. V. Danchev and W. Wm. McGovern. The authors regret for the mistake and would like to thank Prof. Danchev for pointing it out.

In this paper, we investigate the commutativity of a semiprime ring R ad-mitting a generalized derivation F with associated derivation D satisfying any one of theproperties: (i) F ( x ) ◦D ( y ) = [ x;y ], (ii) D ( x ) ◦F ( y ) = F [ x;y ], (iii) D ( x ) ◦F ( y ) = xy , (iv) F ( x ◦ y ) = [ F ( x ) ;y ] + [ D ( y ) ;x ], and (v) F [ x;y ] = F ( x ) ◦ y − D ( y ) ◦ x for all x;y in someappropriate subsets of R . 1. IntroductionThe commutativity of prime rings with derivation was initiated by Posner in[13]. Thereafter, several authors have proved commutativity theorems for primeor semiprime rings admitting automorphisms or derivations which are centralizingor commuting on some appropriate subsets of R (see [1-7,9,10,12,14] where furtherreferences can be found).Throughout this paper, R will represent an associative ring with center Z ( R ).For any x;y ∈ R , the symbol [ x;y ] and ( x ◦ y ) stand for the commutator xy − yx and the anti-commutator xy + yx respectively. A ring R is called a 2-torsion freeif whenever 2

In this paper we give a definition of the Hodge type Laplacian � on a non-commutative manifold which is the smooth dense subalgebra of a C ∗ -algebra. We prove that the Laplacian on a quantum Heisenberg manifold is an elliptic operator in the sense that (� + 1)−1 is compact. In non-commutative geometry, the Chevalley-Eilenberg complex is used to produce a cyclic cocycle in Connes' cyclic cohomology via a cycle over an algebra A where a Lie-group action on A is given. The most important result in this direction is the integrality of the pairing of a cyclic 2-cocycle and Rieffel's projection in the non-commutative torus A� (3, 6, 8). In this paper, using the same framework, we investigate a metric aspect of this complex. In fact, we define "Laplacian" on a non-commutative manifold which is slightly different with the one given in (10) and establish a Hodge- type theorem of the Laplacian on a quantum Heisenberg manifold. While the non-commutative torus is simpler, quantum Heisenberg manifolds with the non-commutative Heisenberg group action are tractable non-commutative man- ifolds given by the stirct deformation quantization of the classical Heisenberg manifold (7). We emphasize that if the group action is commutative and a C ∗ -algebra A is deformed from the group, this is not so interesting since the metric aspect on A is almost commutative as we observe the non-commutative torus case (3, 5). Thus it seems natural to consider the Heisenberg group action and a deformation from it. We show that in this case the Laplacian on zero forms or "functions" is diagonalizable and eigenvalues form a discrete set of R + which is well-known for a connected, oriented Riemmanian manifold (see, for example, (9)).

In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if $(A,\ast )$ is a commutative $d$-algebra and $A^{\mathfrak {d}}$ its Dedekind completion, then, $A^{\mathfrak {d}}$ can be equipped with a $d$-algebra multiplication that extends the multiplication of $A$. \endgraf Moreover, we indicate an error made in the main result of the paper: M. A. Toumi, Extensions of orthosymmetric lattice bimorphisms, Proc. Amer. Math. Soc. 134 (2006), 1615–1621.

The paper is devoted to the problem of describing unary algebras whose congruence lattices have a given property. By now this problem has been solved for algebras with one unary operation. In the paper it is shown that this problem is much more difficult for arbitrary commutative unary algebras. We give some necessary conditions for such lattices to be distributive or modular. Besides, it is proved here that a lattice of all subsets of a set is isomorphic to the congruence lattice of a suitable connected commutative unary algebra.

Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group SUq(2) is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new Toeplitz quantization. Annihilation and creation operators are defined as densely defined Toeplitz operators acting in a quantum Hilbert space, and their commutation relations are discussed. At this point Planck’s constant is introduced into the theory. Due to the possibility of non-commuting symbols, there are now two definitions for anti-Wick quantization; these two definitions are equivalent in the commutative case. The Toeplitz quantization introduced here satisfies one of these definitions, but not necessarily the other. This theory should be considered as a second quantization, since it quantizes non-commutative (that is, already quantum) objects. The quantization theory presented here has two essential features of a physically useful quantization: Planck’s constant and a Hilbert space where natural, densely defined operators act.

In this paper, we introduce the semi-graded rings, which extend graded rings and skew Poincaré–Birkhoff–Witt (PBW) extensions. For this new type of non-commutative rings, we will discuss some basic problems of non-commutative algebraic geometry. In particular, we will prove some elementary properties of the generalized Hilbert series, Hilbert polynomial and Gelfand–Kirillov dimension. We will extend the notion of non-commutative projective scheme to the case of semi-graded rings and we generalize the Serre–Artin–Zhang–Verevkin theorem. Some examples are included at the end of the paper.

In our paper "Non-commutative desingularization of determinantal varieties, I" we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper we prove the existence of non-commutative resolutions in the general case in a manner which is still characteristic free, and carry out the explicit description by generators and relations in characteristic zero. As an application of our results we prove that there is a fully faithful embedding between the bounded derived categories of the two canonical (commutative) resolutions of a determinantal variety, confirming a well-known conjecture of Bondal and Orlov in this special case., Comment: 61 pages, greatly expanded. Now includes a complete treatment of the case of characteristic zero. All comments welcome

Let R be a commutative Noetherian ring and let I be anideal of R . In this paper we study the amalgamated duplication ring R ▷◁ I which is introduced by D’Anna and Fontana. It is shown thatif R is generically Cohen-Macaulay (resp. generically Gorenstein) and I is generically maximal Cohen-Macaulay (resp. generically canonicalmodule), then R ▷◁ I is generically Cohen-Macaulay (resp. genericallyGorenstein). We also de ned generically quasi-Gorenstein ring and weinvestigate when R ▷◁ I is generically quasi-Gorenstein. In addition, itis shown that R ▷◁ I is approximately Cohen-Macaulay if and only if R is approximately Cohen-Macaulay, provided some special conditions.Finally it is shown that if R is approximately Gorenstein, then R ▷◁ I isapproximately Gorenstein. 1. IntroductionThroughout this paper all rings are considered commutative with identityelement and all ring homomorphisms are unital. In [8], D’Anna and Fontanaconsidered a fft type of construction obtained involving a ring

In this paper we use subsets and algebraic substructures of a 3-prime nearring R admitting a derivation d to study the commutativity of the subsets, the algebraic substructures and the near-ring R under suitable conditions on d, R and the algebraic substructures. The results obtained in this paper generalize several commutativity theorems due to some authors.

In this paper, we introduce the category of blueprints, which is a category of algebraic objects that include both commutative (semi)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp. congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and F 1 -schemes (after Kato, Deitmar and Connes–Consani). Beside this unification, the category of blueprints contains new interesting objects as “improved” cyclotomic field extensions F 1 n of F 1 and “archimedean valuation rings”. It also yields a notion of semiring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Titsʼ idea of Chevalley groups over F 1 , congruence schemes, sheaf cohomology, K-theory and a unified view on analytic geometry over F 1 , adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.

This paper is a posthume continuation of the papers by Professor A.H. Boers about duplication of algebras, as drafted by Artibano Micali and Moussa Ouattara in collaboration with Nakelgbamba Boukary Pilabre. The aim of the paper is essentially to study the non commutative duplication of algebras. Conditions are studied in which the duplicate algebra is flexible, Lie admissible or n -associative. Connections between the dimensions of commutative and non commutative duplication of algebras are compared. In the list of references in this paper, several items have been given, which are directly connected to the subject of study. Moreover, for the interest of the reader, a (possible) complete list of publications of A.H. Boers is added to the references, i.e. known up to and including 1995 as well as a note explicative.

As a new generalization of the notion of the standard group, the notion of the neutrosophic triplet group (NTG) is derived from the basic idea of the neutrosophic set and can be regarded as a mathematical structure describing generalized symmetry. In this paper, the properties and structural features of NTG are studied in depth by using theoretical analysis and software calculations (in fact, some important examples in the paper are calculated and verified by mathematics software, but the related programs are omitted). The main results are obtained as follows: (1) by constructing counterexamples, some mistakes in the some literatures are pointed out, (2) some new properties of NTGs are obtained, and it is proved that every element has unique neutral element in any neutrosophic triplet group, (3) the notions of NT-subgroups, strong NT-subgroups, and weak commutative neutrosophic triplet groups (WCNTGs) are introduced, the quotient structures are constructed by strong NT-subgroups, and a homomorphism theorem is proved in weak commutative neutrosophic triplet groups.

In this paper, one kind of subgroup in arbitrary group which similar to the linear subspace was constructed, and the generalization of the Hahn-Banach theorem on this kind of subgroup in arbitrary groups was obtained. The Hahn-Banach theorem is a powerful existence theorem whose form is par- ticularly appropriate to applications in linear problems. In its elegance and power, the Hahn-Banach theorem is a favorite of almost every analyst. The generalization of the Hahn-Banach theorem on groups has been discussed in many articles, much of these discussions were under the assumption of some condition of groups, such as the weakly commutativity in the paper of Z. Gajda and Z. Kominek (3), or groups in class G during R. Badora (1). The purpose of this paper is to find the sucient and necessary condition of Hahn-Banach theorem on arbitrary groups. Let G be a group, p,f be functionals on G ! R, then p is subadditive and f is additive if and only if p(xy) p(x) + p(y), x,y 2 G and f(xy) = f(x) + f(y), x,y 2 G. Moreover, p is completely commutative if and only if for any n 2 Z + and any per- mutation xk1,··· ,xkn of x1,··· ,xn 2 G, one has p( n Y i=1 xi) = p( n Y i=1 xki ).

Let R be a quotient ring of a commutative coherent regular ring by a finitely generated ideal. Hovey gave a bijection between the set of coherent subcategories of the category of finitely presented R-modules and the set of thick subcategories of the derived category of perfect R-complexes. Using this isomorphism, he proved that every coherent subcategory of finitely presented R-modules is a Serre subcategory. In this paper, it is proved that this holds whenever R is a commutative noetherian ring. This paper also yields a module version of the bijection between the set of localizing subcategories of the derived category of R-modules and the set of subsets of Spec R which was given by Neeman., 17 pages, to appear in J. Lond. Math. Soc

Let A and B be semisimple commutative Banach algebras with bounded approximate identities. We investigate the problem of extending a homomorphism ' : A ! B to a homomorphism of the multiplier algebras M(A) and M(B) of A and B, respectively. Various sufficient conditions in terms ofB (or B and ') are given that allow the construction of such extensions. We exhibit a number of classes of Banach algebras to which these criteria apply. In addition, we prove a polar decomposition for homomorphisms from A into A with closed range. Our results are applied to Fourier algebras of locally compact groups. Introduction. Let A and B be semisimple commutative Banach alge- bras and suppose that A has a bounded approximate identity (e�)�. We study homomorphisms ϕ from A to B from various aspects. Let I' be the largest ideal of B for which (ϕ(e�))� serves as an approximate identity, and let Z' denote the zero set of I' in the Gelfand spectrum � (B) of B. In Section 1 we find criteria for Z' to be open in � (B) and I' to be comple- mented by a certain ideal J' (Theorems 1.4 and 1.5). The results are applied to the related question of when a homomorphism ϕ : A → B extends to a homomorphism, φ : M(A) → M(B), between the multiplier algebras M(A) and M(B). This extension problem is the main objective of the paper. When I' is complemented as above, we give in Theorem 2.1 (which is a basic result of the paper) an explicit construction of an extension φ : M(A) → M(B). Moreover, if in addition B is a BSE-algebra (named after Bochner-Schoenberg-Eberlein), then all homomorphisms from M(A)

In this paper we establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (gaw) holds. In this work, we consider commutative perturbations by algebraic operator and quasinilpotent operator for T ∈ B(X ) such that T * satisfies property (gaw). We prove that if A is an algebraic and T ∈ PS(X ) is such that AT = TA, then ƒ(T * + A*) satisfies property (gaw) for every ƒ ∈ Hc(σ(T + A)). Moreover, we show that if Q is a quasi-nilpotent operator and T ∈ PS(X ) is such that TQ = QT, then ƒ(T * + Q*) satisfies the property (gaw) for every ƒ ∈ Hc(σ(T +Q)). At the end of this paper, we apply the obtained results to a number of subclasses of PS(X ).

The paper consists of two parts. The first part introduces the representation ring for the family of compact unitary groups U(1), U(2),.... This novel object is a commutative graded algebra R with infinite-dimensional homogeneous components. It plays the role of the algebra of symmetric functions, which serves as the representation ring for the family of finite symmetric groups. The purpose of the first part is to elaborate on the basic definitions and prepare the ground for the construction of the second part of the paper. The second part deals with a family of Markov processes on the dual object to the infinite-dimensional unitary group U(infinity). These processes were defined in a joint work with Alexei Borodin (J. Funct. Anal. 2012; arXiv:1009.2029). The main result of the present paper consists in the derivation of an explicit expression for their infinitesimal generators. It is shown that the generators are implemented by certain second order partial differential operators with countably many variables, initially defined as operators on R., Comment: version 2: 67 pp.; minor improvements, typos fixed; version 3: minor typographical corrections, two uncited references deleted; to appear in Adv. Math