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We show that the endomorphism ring of any nonzero finitely generated projective module over the Leavitt path algebra $L_K(E)$ of an arbitrary graph $E$ with coefficients in a field $K$ is isomorphic to a Steinberg algebra. This yields in particular that every nonzero corner of the Leavitt path algebra of an arbitrary graph is isomorphic to a Steinberg algebra. This in its turn gives that every $K$-algebra with local units which is Morita equivalent to the Leavitt path algebra of a row-countable graph is isomorphic to a Steinberg algebra. Moreover, we prove that a corner by a projection of a $C^*$-algebra of a countable graph is isomorphic to the $C^*$-algebra of an ample groupoid., Comment: 31 pages

Let K be a field (finite or infinite) of char ( K ) ≠ 2 and let U T 2 ( K ) be the 2 × 2 upper triangular matrix algebra over K. If ⋅ is the usual product on U T 2 ( K ) then with the new product a ∘ b = ( 1 / 2 ) ( a ⋅ b + b ⋅ a ) we have that U T 2 ( K ) is a Jordan algebra, denoted by U J 2 = U J 2 ( K ) . In this paper, we describe the set I of all polynomial identities of U J 2 and a linear basis for the corresponding relatively free algebra. Moreover, if K is infinite we prove that I has the Specht property. In other words I, and every T-ideal containing I, is finitely generated as a T-ideal.

Let L be a finitely generated nilpotent Lie algebra over a field K and let d be the smallest integer such that L can be generated by d elements. Let n ≥ d be a positive integer and suppose that every proper subalgebra of L has class at most n. It is not difficult to show that the class of L is at most n + q where q = ⌊ n / ( d − 1 ) ⌋ . Our main result shows that there exist such Lie algebras of class (exactly) n + q whenever q ≥ 3 and K has characteristic 0 or prime characteristic p such that p does not divide ( q 2 − 1 ) q / 2 .

The aim of this paper is to describe the structure of the finitely generated subgroups of a family of branch groups, which includes the first Grigorchuk group and the Gupta-Sidki 3-group. This description is made via the notion of block subgroup. We then use this to show that all groups in the above family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups., 22 pages, 2 figures; V3: Final version, to appear in Izvestiya: Mathematics

The purpose of this paper is to prove that the symbolic Rees rings of ideals defining certain finite sets of points in the projective plane over an algebraically closed field are finitely generated using a ring theoretical criterion which is known as Huneke's criterion.

Let H be an affine quasi-hereditary algebra defined in [9] which satisfies the condition given in [6] , that is, H is finitely generated over its center. We prove that the Ringel dual R of H is affine quasi-hereditary when H satisfies certain additional condition. Under the same condition, we prove that the double Ringel dual R R of H is graded Morita equivalent to H. In particular, if all the irreducible H-modules have dimension 1, then R R ≅ H as graded algebras.

In this paper, we introduce a new class of modules satisfying S-dccr (S-dccr $$^{\star })$$ condition which is a generalization of S-artinian modules. Let $$A\ $$ be a commutative ring with $$0\ne 1\ $$ and $$X\ $$ a unital A-module. Suppose that $$S\subseteq A\ $$ is a multiplicatively closed subset. $$X\ $$ is said to satisfy S-dccr (S-dccr $$^{\star })$$ condition if for each finitely generated (principal) ideal $$I\ $$ of $$A\ $$ and a submodule $$Y\ $$ of $$X,\ $$ the descending chain $$\{I^{i}Y\}_{i\in {\mathbb {N}}}$$ is S-stationary. Many examples and properties of modules satisfying S-dccr (S-dccr $$^{\star })\ $$ condition are given. Furthermore, we characterize modules satisfying dccr (dccr $$^{\star })$$ condition in terms of some known class of rings and modules. Also, we give Nakayama’s Lemma for modules satisfying S-dccr condition.

We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a linear stable range. This was originally proved for the second terms by Ershov and He., Comment: 37 pages, 3 figures. Major revision (especially to Section 5), to appear in JEMS

We study systematically groups whose marked finite quotients form a recursive set. We give several definitions, and prove basic properties of this class of groups, and in particular emphasize the link between the growth of the depth function and solvability of the word problem. We give examples of infinitely presented groups whose finite quotients can be effectively enumerated. Finally, our main result is that a residually finite group can be even not recursively presented and still have computable finite quotients, and that, on the other hand, it can have solvable word problem while still not having computable finite quotients., Comment: 23 pages, 0 figure. Improved version, Problem 12 was in fact already solved

An integral domain $D,$ with quotient field $K,$ is a $v$-domain if for each nonzero finitely generated ideal $A$ of $D$ we have $(AA^{-1})^{-1}=D.$ It is well known that if $D$ is a $v$-domain$,$ then some quotient ring $D_{S}$ of $D$ may not be a $v$-domain. Calling $D$ a super $v$-domain if every quotient ring of $D$ is a $v$-domain we characterize super $v$-domains as locally $v$-domains. Using techniques from factorization theory we show that $D$ is a super $v$-domain if and only if $D[X]$ is a super $v$-domain if and only if $D+XK[X]$ is a super $v$-domain and give new examples of super $v$ -domains that are strictly between $v$-domains and P-domains that were studied in [Manuscripta Math. 35(1981)1-26]

We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism., Comment: 10 pages, 2 figures

For an ascending HNN-extension $G*_{\psi}$ of a finitely generated abelian group $G$, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal{A}^{G*_{\psi}}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag-Solitar groups $\mathrm{BS}(1,N)$, $N\ge2$, for which our results imply that a $\mathrm{BS}(1,N)$-SFT which contains a configuration with period $a^{N^\ell}$, $\ell\ge 0$, must contain a strongly periodic configuration with monochromatic $\mathbb{Z}$-sections. Then we study proper $n$-colorings, $n\ge 3$, of the (right) Cayley graph of $\mathrm{BS}(1,N)$, estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm{BS}(1,N)$ admits a frozen $n$-coloring if and only if $n=3$. We finally suggest generalizations of the latter results to $n$-colorings of ascending HNN-extensions of finitely generated abelian groups., Comment: 22 pages

We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups, if two equidistributed measurable sets are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Mathe and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.

We investigate the relations between Pesin-Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions. Let (X, $${\cal G}$$ ) be a system, where X is a compact metric space and $${\cal G}$$ is a finite family of continuous maps on X. Given a continuous function f on X, we define Pesin-Pitskel topological pressure $${P_{\cal G}}(Z,f)$$ for any subset Z ⊂ X and measure-theoretical pressure $${P_{\mu ,{\cal G}}}(X,f)$$ for any $$\mu \in {\cal M}(X)$$ , where $${\cal M}(X)$$ denotes the set of all Borel probability measures on X. For any non-empty compact subset Z of X, we show that $${P_{\cal G}}(Z,f) = \sup \{ {P_{\mu ,{\cal G}}}(X,f):\mu \in {\cal M}(X),\mu (Z) = 1\} .$$

It is proved that the solvability problem for a finite independent system of equations in a finitely generated nilpotent group can effectively be reduced to a similar problem in some finite quotient group of this group. Therefore, this problem is algorithmically solvable. This strengthens a theorem of A. G. Makanin on the residual finiteness and algorithmic solvability of a regular splittable equation in a finitely generated nilpotent group.

Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$, $N$ be two Cohen-Macaulay $A$-modules with $M$ linked to $N$ via a Gorenstein ideal $\mathfrak{q}$. Let $L$ be another finitely generated $A$-module. We show that $\mathrm{Ext}^i_A(L,M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Tor}^A_i(L,N) = 0$ for all $i \gg 0$. If $D$ is a Cohen-Macaulay module then we show that $\mathrm{Ext}^i_A(M, D) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(D^\dagger , N) = 0$ for all $i \gg 0$, where $D^\dagger = \mathrm{Ext}^r_A(D,A)$ and $r = \mathrm{codim}(D)$. As a consequence we get that $\mathrm{Ext}^i_A(M, M) = 0 $ for all $i \gg 0$ if and only if $\mathrm{Ext}^i_A(N, N) = 0$ for all $i \gg 0$. We also show that $\mathrm{End}_A(M)/\mathrm{rad}\,\mathrm{End}_A(M) \cong (\mathrm{End}_A(N)/\mathrm{rad}\,\mathrm{End}_A(N))^{\mathrm{op}}$. We also give a negative answer to a question of Martsinkovsky and Strooker.

In this article, we analyze the category G-R-mod of unitary G-graded modules over object unital G-graded rings R, being G a groupoid. Here we consider the forgetful functor U:G-R-mod→R-mod and dete...

Let K be a number field, and let G be a finitely generated subgroup of $$K^\times $$ . We are interested in computing the degree of the cyclotomic-Kummer extension $$K(\root n \of {G})$$ over K, where $$\root n \of {G}$$ consists of all n-th roots of the elements of G. We develop the theory of entanglements introduced by Lenstra, and we apply it to compute the above degrees.

In this article, we classify the irreducible integrable modules for the loop affine-Virasoro algebra ((g°⊗C[t,t−1]⊕CK)⋊ Vir)⊗A, where A is a finitely generated commutative associative unital algebr...

A module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that $$V \subseteq U$$ V ⊆ U and M/V is finitely generated. It is shown that every non-finitely generated free module is $$\mathfrak {s}$$ s -coseparable but a finitely generated free module is not, in general, $$\mathfrak {s}$$ s -coseparable. We prove that the class of $$\mathfrak {s}$$ s -coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is $$\mathfrak {s}$$ s -coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is $$\mathfrak {s}$$ s -coseparable are provided.

We introduce various new type of recurrent sets for finitely generated semigroups on non-compact metric spaces that are conjugacy invariant, and obtain some basic properties of chain recurrent sets for semigroups via these new definitions. Moreover, we define the notion of weak shadowing property for finitely generated group actions on compact metric spaces, which is weaker than that of shadowing property, and prove the equivalence of the shadowing and weak shadowing properties for the finitely generated group actions on a generalized homogeneous space without isolated points.

A concept of Cohen–Macaulay in codimension t is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied CMt simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen–Macaulay and Buchsbaum properties. The purpose of this paper is to survey briefly recent results of CMt simplicial complexes.

Let [Formula: see text] be a finitely generated field over [Formula: see text] and fix [Formula: see text]. We study the solutions of the Catalan equation [Formula: see text] to be solved in [Formula: see text] and integers [Formula: see text] coprime with [Formula: see text]. Our main result corrects earlier work of Silverman.

The critical relations are the building blocks of the relational clone of a relational structure with respect to the relational operations intersection and direct product. In this paper we describe the critical relations of crowns. As a consequence, we obtain that the subpower membership problem for any crown is polynomial-time solvable.

Clone theory plays an important role in studying aggregation functions on bounded posets or bounded lattices. Several important classes of aggregation functions on a bounded lattice L form a clone, particularly the set of all aggregation functions on L, the so-called full aggregation clone on L. For any finite lattice L, this clone is known to be finitely generated and various generating sets and their constructions have been presented in recent papers. The aim of this paper is to extend previous results concerning generating sets of aggregation clones on finite chains. Namely, the objective is to discuss the minimality of certain generating bases, the so-called ( χ , ⊕ ) -generating sets.

Infinitely generated closed classes $$\Pi_{1}{-}\Pi_{4}$$ of 01-functions of three-valued logic are investigated. The property of maximality is proved for classes $$\Pi_{3}$$ and $$\Pi_{4}$$ . Each proper extension of classes leads to finitely generated closed classes. It is proved that there is no base in class $$\Pi_{1}$$ . All of the simplest two- and three-variable functions obtainable by superposition from an arbitrary function not belonging to class $$\Pi_{1}$$ are found.

For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb{Q}^{\times}$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb{Q}$ and a union of conjugacy classes $C \subset \text{Gal}(F/\mathbb{Q})$, we develop methods for determining if there exists infinitely many primes $p$ such that the index of the reduction of $W_i$ modulo $p$ divides $k_i$ and such that the Artin symbol of $p$ on $F$ is contained in $C$. The results are a multivariable generalization of H.W. Lenstra's work. As an application, we determine all integers $a_1, \ldots , a_n$ such that $\text{ord}_p(a_1) = \ldots = \text{ord}_p(a_n)$ for infinitely many primes $p$. We also discuss the set of those $p$ for which $\text{ord}_p(a_1) > \ldots > \text{ord}_p(a_n)$. The obtained results are conditional to a generalization of the Riemann hypothesis., 20 pages. Add section on Kummer-type extensions and improve exposition

We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple gluing of such. In one of our simplest constructions—the pocket-extension of a group G—this leads to the study of certain finitely generated subgroups of the full permutation group $${\mathbb {S}}(G\cup \{*\})$$ . Some sharp estimates are obtained while many challenging questions remain.

Let $\Lambda$ be an artin algebra and let $\mathcal{P}^{, Comment: 4 pages

The aim of this paper is to study the Hopfian property in the context of chain and duo rings. For such rings, we characterize Hopfian free modules and show that a direct sum of cyclic R-modules is Hopfian if and only if the sum is finite. This allows us to show that finitely generated modules over a local right duo ring, which has the FGC-property, are Hopfian and cancel in direct sums. Moreover, being finitely, hopficity, and the cancellation property are equivalent for modules over Artinian rings.

Every Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we strengthen and refine this tool for Krull monoids with finitely generated class group.

In 2004, C. Sanza, with the purpose of legitimizing the study of \(n\times m\)-valued Łukasiewicz algebras with negation (or \(\mathbf{NS}_{n\times m}\)-algebras) introduced \(3 \times 3\)-valued Łukasiewicz algebras with negation. Despite the various results obtained about \(\mathbf{NS}_{n\times m}\)-algebras, the structure of the free algebras for this variety has not been determined yet. She only obtained a bound for their cardinal number with a finite number of free generators. In this note we describe the structure of the free finitely generated \(NS_{3 \times 3}\)-algebras and we determine a formula to calculate its cardinal number in terms of the number of free generators. Moreover, we obtain the lattice \(\Lambda(\mathbf{NS}_{3\times 3})\) of all subvarieties of \(\mathbf{NS}_{3\times 3}\) and we show that the varieties of Boolean algebras, three-valued Łukasiewicz algebras and four-valued Łukasiewicz algebras are proper subvarieties of \(\mathbf{NS}_{3\times 3}\).

It is a classical result that the direct product $$A\times B$$ of two groups is finitely generated (finitely presented) if and only if A and B are both finitely generated (finitely presented). This is also true for direct products of monoids, but not for semigroups. The typical (counter) example is when A and B are both the additive semigroup $${\mathbb {P}}=\{1,2,3,\ldots \}$$ of positive integers. Here $${\mathbb {P}}$$ is freely generated by a single element, but $${\mathbb {P}}^2$$ is not finitely generated, and hence not finitely presented. In this note we give an explicit presentation for $${\mathbb {P}}^2$$ in terms of the unique minimal generating set; in fact, we do this more generally for $${\mathbb {P}}^K$$ , the direct product of arbitrarily many copies of $${\mathbb {P}}$$ .

We show, among other things, that for each integer $$n \ge 3$$ , there is a smooth complex projective rational variety of dimension n, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.

Let K be a finitely generated extension of a field k of characteristic $$p\not =0$$ . In 1947, Dieudonne initiated the study of maximal separable intermediate fields. He gave in particular the form of an important subclass of maximal separable intermediate fields D characterized by the property $$K\subseteq k({D}^{p^{-\infty }})$$ , and which are called the distinguished subfields of K/k. In 1970, Kraft showed that the distinguished maximal separable subfields are precisely those over which K is of minimal degree. This paper grew out of an attempt to find a new characterization of distinguished subfields of K/k by means of new inseparability invariants.

Let [Formula: see text] and [Formula: see text] be artin algebras and [Formula: see text] the triangular matrix algebra with [Formula: see text] a finitely generated ([Formula: see text])-bimodule. We construct support [Formula: see text]-tilting modules and ([Formula: see text]-)tilting modules in [Formula: see text] from that in [Formula: see text] and [Formula: see text], and give the converse constructions under some condition.

Given two non-empty subsets $W,W'\subseteq G$ in an arbitrary abelian group $G$, $W'$ is said to be an additive complement to $W$ if $W + W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $W$. The notion was introduced by Nathanson and previous work by him, Chen--Yang, Kiss--S\`andor--Yang etc. focussed on $G =\mathbb{Z}$. In the higher rank case, recent work by the authors treated a class of subsets, namely the eventually periodic sets. However, for infinite subsets, not of the above type, the question of existence or inexistence of minimal complements is open. In this article, we study subsets which are not eventually periodic. We introduce the notion of "spiked subsets" and give necessary and sufficient conditions for the existence of minimal complements for them. This provides a partial answer to a problem of Nathanson., Comment: 25 pages, 8 figures

We demonstrate the existence of a family of finitely generated subgroups of Richard Thompson's group $F$ which is strictly well-ordered by the embeddability relation in type $\epsilon_0 +1$. All except the maximum element of this family (which is $F$ itself) are elementary amenable groups. In fact we also obtain, for each $\alpha < \epsilon_0$, a finitely generated elementary amenable subgroup of $F$ whose EA-class is $\alpha + 2$. These groups all have simple, explicit descriptions and can be viewed as a natural continuation of the progression which starts with $\mathbf{Z} + \mathbf{Z}$, $\mathbf{Z} \wr \mathbf{Z}$, and the Brin-Navas group $B$. We also give an example of a pair of finitely generated elementary amenable subgroups of $F$ with the property that neither is embeddable into the other., Comment: 47 pages. Substantially revised, with heaviest revisions in sections 8 and 9. Accepted for publication in Journal of Combinatorial Algebra

Abstract We are studying the punctual structures, i.e., the primitive recursive structures on the whole set of integers. The punctual categoricity relative to a computable oracle $$f$$ means that between any two punctual copies of a structure there is an isomorphism which togeteher with its inverse can be derived via primitive recursive schemes augmented with $$f$$. We will prove that the punctual categoricity relative to a computable oracle can hold only for finitely generated or locally finite structures. We will show that the punctual categoricity of finitely generated structures is exhaused by the computable oracles with primitive recursive graph. We also present an example of locally finite structure where the punctual categoricity is provided by a primitive recursively bounded computable oracle.

Let R be a commutative Noetherian local ring and M , N be finitely generated R-modules. We prove a number of results of the form: if Hom R ( M , N ) has some nice properties and Ext R 1 ≤ i ≤ n ( M , N ) = 0 for some n, then M (and sometimes N) must be close to free. Our methods are quite elementary, yet they suffice to give a unified treatment, simplify, and sometimes extend a number of results in the literature.

Let S be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if R is a non-Gorenstein quotient of S of small colength, then every totally reflexive R-module is free. Indeed, the second syzygy of the canonical module of R has a direct summand T which is a test module for freeness over R in the sense that if Tor + R ( T , N ) = 0 , for some finitely generated R-module N, then N is free.

Suppose a finitely generated group $G$ is hyperbolic relative to $\mathcal P$ a set of proper finitely generated subgroups of $G$. Established results in the literature imply that a "visual" metric on $\partial (G,\mathcal P)$ is "linearly connected" if and only if the boundary $\partial (G,\mathcal P)$ has no cut point. Our goal is to produce linearly connected metrics on $\partial (G,\mathcal P)$ that are "piecewise" visual when $\partial (G,\mathcal P)$ contains cut points. %Visual metrics for $\partial (G,\mathcal P)$ are tightly linked to inner products of geodesic rays in "cusped" spaces for $(G,\mathcal P)$. The identity vertex $\ast$ is usually our base point in these cusped spaces and visual metrics depend on this base point. %We say the visual metric $d_p$ on $\partial(G,\mathcal P)$, with base point $p$, is {\it $G$-equivariant} if for points $x_1,x_2\in \partial(G,\mathcal P)$, we have $d_p(x_1,x_2)=d_{gp}(gx_1,gx_2)$ for all $g\in G$. Our main theorem is about graph of groups decompositions of relatively hyperbolic groups $(G,\mathcal P)$, and piecewise visual metrics on their boundaries. We assume that each vertex group of our decomposition has a boundary with linearly connected visual metric or the vertex group is in $\mathcal P$. If a vertex group is not in $\mathcal P$, then it is hyperbolic relative to its adjacent edge groups. Our linearly connected metric on $\partial (G,\mathcal P)$ agrees with the visual metric on limit sets of vertex groups and is in this sense piecewise visual., 50 pages, 16 figures

The ring Int(Z) of integer-valued polynomials has the two-generator property, which means that every finitely generated ideal may be generated by two elements. As the known proofs of this fact are ...

Cho (R, m) là vành giao hoán Noether và Q(R) là vành các thương toàn phần của R. Mục đích của bài báo này là nghiên cứu cấu trúc của các vành trung gian giữa R và Q(R). Gọi X là tập tất cả các lớp tương đương [I], trong đó I là ideal của R sao cho I 2 = aI với a ∈ I là phần tử không là ước của không trong R. Gọi Y là tập tất cả các vành trung gian A giữa R và Q(R) sao cho A là R-môđun hữu hạn sinh. Trong bài báo này, chúng tôi thiết lập một song ánh từ X đến Y. Một số ví dụ được đưa ra để làm rõ kết quả. Thứ nhất, chúng tôi chỉ ra nếu R là một miền ideal chính thì R là phần tử duy nhất của Y. Thứ hai, cho một vành Buchsbaum R mà không là Cohen-Macaulay, chúng tôi xây dựng một vành trung gian Cohen-Macaulay A ∈ Y. Để giải quyết vấn đề, chúng tôi áp dụng phương pháp nghiên cứu của S. Goto năm 1983, L. T. Nhàn và M. Brodmann 2012.

Section 1 gives a brief review of known results on embeddings of solvable, nilpotent, and polycyclic groups in 2-generated groups from these classes, including the description of the author’s recently obtained solution to the Mikaelian–Ol’schanskii problem on embeddings of finitely generated solvable groups of derived length l in solvable groups of derived length l + 1 with a fixed small number of generators. Section 2 contains a somewhat more extensive review of known results on the rational subset membership problem for groups, including the presentation of the author’s recently obtained solution to the Laurie–Steinberg–Kambites–Silva–Zetsche problem of whether the membership problem is decidable for finitely generated submonoids of free nilpotent groups.