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A question of Conca, Rossi, and Valla asks whether every quadratic Gorenstein ring $R$ of regularity three is Koszul. In a previous paper, we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three which are not Koszul. In this paper, we study the analog of the Conca-Rossi-Valla question when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein ring having $\mathrm{codim}\, R = c$ and $\mathrm{reg}\, R = r \ge 4$. We prove that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r+2$, we construct quadratic Gorenstein rings that are not Koszul, answering questions of Matsuda and Migliore-Nagel concerning the $h$-vectors of quadratic Gorenstein rings., Comment: v2 - Minor changes based on referee comments

In the present paper, the notion of (additive) local multiplier on the ring of matrices over an arbitrary unital associative ring is introduced and investigated. It is proved that every local left multiplier on the matrix ring over a division ring is a left multiplier. Also, in the present paper, the notion of (additive) local Jordan multiplier on a Jordan ring of symmetric matrices is introduced and investigated. It is proved that every local Jordan multiplier on the Jordan ring of symmetric matrices over the field of rational numbers is a Jordan multiplier. Also corollaries of these results are proved.

The aim of this paper is to present a Grobner-Shirshov basis for a special type of braid monoids, namely the symmetric inverse monoid In, in terms of the dex-leg ordering on the related elements of monoid. By taking into account the Grobner-Shirshov basis, the ideal form (or, equivalently, the normal form structure) of this important monoid will be obtained. This ideal form will give us the solution of the word problem. At the final part of this paper, we give an application of our main result which find out a Grobner-Shirshov basis for the symmetric inverse monoid I4 such that the accuracy and efficiency of this example can be seen by GBNP package in GAP (Group, Algorithms and Programming) which computes Grobner bases of non-commutative polynomials [ 1 ].

Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\Gamma$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $\Gamma$. We also investigate the situation where both $A,\Gamma$ have finite global dimension. When $A$ is Koszul and finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determiant of artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $\Gamma$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture., Comment: 14 pages

In paper is considered the development of the concept of "artinian"for Lie algebras. The concept of artinian was introduced for associative rings with the minimality condition. At the same time, it extended to modules and subalgebras. A little later they began to consider Artinian Jordan algebras. For such algebras the role of a one-sided ideal is played by a quadratic ideal or, as N. Djecobson called it, the inner ideal. Artinian for Lie algebras through ideals determined Yu.A. Bakhturin, S.A. Pikhtilkov and V.М. Polyakov. They considered special Artinian Lie algebras. S.A. Pikhtilkov applied Artinian Lie algebras to construct the structural theory of special Lie algebras. Georgia Benkart defined the artinian for Lie algebras through inner ideals. F. Lopez, E. Garcia, G. Lozano explored the concept of the inner ideal applied to artinian with the help of Jordan pairs. The definition of artinian for Lie algebras in this paper is presented in three senses: via subalgebras, ideals, and inner ideals. The relationship established between these definitions is established by the authors earlier. Examples of Artinian Lie algebras are considered. The application of Artinian Lie algebras to the solution of the Mikhalev problem is described: the primary radical of the Artinian Lie algebra is solvable.

The notion of π-regular endomorphism ring of an abelian group, which generalizes the notion of regular endomorphism ring, was introduced in papers of L. Fuchs and K. Rangaswamy. They described periodic abelian groups with π-regular endomorphism ring and found necessary conditions for an abelian group to have π-regular endomorphism ring. In this paper, we study abelian groups with sufficiently π-regular endomorphism ring, which form a subclass of the class of abelian groups with π-regular endomorphism ring, and find necessary and sufficient conditions for an abelian group to have sufficiently π-regular endomorphism ring.

We raise the following general question regarding a ring graded by a group: "If $P$ is a ring-theoretic property, how does one define the graded version $P_{\operatorname{gr}}$ of the property $P$ in a meaningful way?". Some properties of rings have straightforward and unambiguous generalizations to their graded versions and these generalizations satisfy all the matching properties of the nongraded case. If $P$ is either being unit-regular, having stable range 1 or being directly finite, that is not the case. The first part of the paper addresses this issue. Searching for appropriate generalizations, we consider graded versions of cancellation, internal cancellation, substitution, and module-theoretic direct finiteness. In the second part of the paper, we turn to Leavitt path algebras. If $K$ is a trivially graded field and $E$ is an oriented graph, the Leavitt path algebra $L_K(E)$ is naturally graded by the ring of integers. If $E$ is a finite graph, we present a property of $E$ which is equivalent with $L_K(E)$ being graded unit-regular. This property critically depends on the lengths of paths to cycles making it stand out from other known graph conditions which characterize algebraic properties of $L_K(E).$ It also further illustrates that graded unit-regularity is quite restrictive in comparison to the alternative generalization of unit-regularity which we consider in the first part of the paper.

A theorem due to Nakayama and Skornyakov states that “a ring R is an Artinian serial ring if and only if all left R-modules are serial” and a theorem due to Warfield state that “a Noetherian ring R is serial if and only if every finitely generated left R-module is serial”. We say that an R-module M is prime uniserial (℘-uniserial, for short) if for every pair P, Q of prime submodules of M either \(P\subseteq Q\) or \(Q\subseteq P\), and we say that M is prime serial (℘-serial, for short) if it is a direct sum of ℘-uniserial modules. Therefore, two interesting natural questions of this sort are: “Which rings have the property that every module is ℘-serial?” and “Which rings have the property that every finitely generated module is ℘-serial?” Most recently, in our paper, Prime uniserial modules and rings (submitted), we considered these questions in the context of commutative rings. The goal of this paper is to answer these questions in the case R is a Noetherian ring in which all idempotents are central or R is a left Artinian ring.

In this paper we study the notion of a graded $\Omega$-group $(X,+,\Omega),$ but graded in the sense of M.~Krasner, i.e., we impose nothing on the grading set except that it is nonempty, since operations of $\Omega$ and the grading of $(X,+)$ induce operations (generally partial) on the grading set. We prove that graded $\Omega$-groups in Krasner's sense are determined up to isomorphism by their homogeneous parts, which, with respect to induced operations, represent partial structures called $\Omega$-\emph{homogroupoids}, thus narrowing down the theory of graded $\Omega$-groups to the theory of $\Omega$-homogroupoids. This approach already proved to be useful in questions regarding A. V. Kelarev's $S$-graded rings inducing $S,$ where $S$ is a partial cancellative groupoid. Particularly, in this paper we prove that the homogeneous subring of a Jacobson $S$-graded ring inducing $S$ is Jacobson under certain assumptions. We also discuss the theory of prime radicals for $\Omega$-homogroupoids thus extending results of A.~V.~Mikhalev, I.~N.~Balaba and S.~A.~Pikhtilkov in a natural way. We study some classes of $\Omega$-homogroupoids for which the lower and upper weakly solvable radicals coincide and also, study the question of the homogeneity of the prime radical of a graded ring.

In this paper we consider Artin local K-algebras with maximal length in the class of Artin algebras with given embedding dimension and socle type. They have been widely studied by several authors, among others by Iarrobino, Froberg and Laksov. If the local K-algebra is Gorenstein of socle degree 3, then the authors proved that it is canonically graded, i.e. analytically isomorphic to its associated graded ring, see (6). This unexpected result has been extended to compressed level K-algebras of socle degree 3 in (4). In this paper we end the investigation proving that extremal Artin Gorenstein local K-algebras of socle degree s � 4 are canonically graded, but the result does not extend to extremal Artin Gorenstein local rings of socle degree 5 or to compressed level local rings of socle degree 4 and type > 1. As a consequence we present results on Artin compressed local K-algebras having a specified socle type.

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures., 15 pages, to appear in European Journal of Mathematics

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

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 this paper, we define a notion of AS-Gorenstein algebra for N -graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.

For a ring endomorphism of a ring R , Krempa called a rigid endomorphism if a ( a ) = 0 implies a = 0 for a 2 R , and Hong etal. called R an - rigid ring if there exists a rigid endomorphism . Dueto Rege and Chhawchharia, a ring R is called Armendariz if wheneverthe product of any two polynomials in R [ x ] over R is zero, then so isthe product of any pair of coffits from the two polynomials. TheArmendariz property of polynomials was extended to one of skew poly-nomials (i.e., -Armendariz rings and -skew Armendariz rings) by Honget al. In this paper, we study the relationship between -rigid rings andextended Armendariz rings, and so we get various conditions on the ringswhich are equivalent to the condition of being an -rigid ring. Severalknown results relating to extended Armendariz rings can be obtained ascorollaries of our results. Throughout this paper, all rings are associative with identity. Given a ring R , the polynomial ring over R is denoted by R [ x ]. Recall that a ring

The aim of this paper is to study the correlation between the properties of rings and the structure of their additive groups. In this paper, all multiplications on reduced algebraically compact groups and on divisible torsion-free Abelian groups are described. Absolute Jacobson radicals and absolute nil-radicals in some classes of Abelian groups are found. Using these results, we will give a description of semisimple groups and also of groups on which every ring is a nilpotent ring (a nil-ring, a radical ring) in some concerned classes of Abelian groups.

In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables., 20 pages. This paper generalizes a previous version where the result was proven for a power series ring in two variables

Let R be an associative ring with unit. A nonsemisimple right R-module M = MR is referred to as a (right) Schmidt module if every proper (right) submodule in M is semisimple, and a module M is called a (right) generalized Schmidt module if M is not a Schmidt module and each of its proper (right) submodule is either a semisimple module or a Schmidt module. A left Schmidt R-module and a left generalized Schmidt R-module are defined similarly. In the paper, a complete description of the structure of right Schmidt R-modules and generalized Schmidt R-modules is given, the existence of Schmidt R-submodules in any nonsemisimple Artinian module is established, and a complete description of nonsemisimple Artinian modules in which every Schmidt submodule is distinguished as a direct summand is presented. As corollaries, characterizations of (generalized) Schmidt modules over a Dedekind ring and over a matrix ring over this ring are obtained in the paper.

Since the theory of ideals plays an important role in the theory of quotient semirings, in this paper, we will make an intensive study of the notions of Noetherian, Artinian, prime, primary, weakly primary and k-maximal ideals in commutative quotient semirings. The bulk of this paper is devoted to stating and proving analogues to several well-known theorems in the theory of the residue class rings.

In a previous paper, Auslander–Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincare duality space, each component of the Auslander–Reiten quiver is isomorphic to \(\mathbb{Z}A_{\infty }\).

In the preceding paper, the authors published an existence theorem for basic submodules of right modules over right Noetherian, serial rings. The aim of the present paper is to prove a uniqueness theorem for basic submodules over such rings. Bibliography: 13 titles.

The paper is a continuation of the authors' study of quasi-hereditary algebras whose Yoneda extension algebras (homological duals) are quasi-hereditary. The so-called standard Koszul quasi-hereditary algebras, presented in this paper, have the property that their extension algebras are always quasi-hereditary. In the natural setting of graded Koszul algebras, the converse also holds: if the extension algebra of a graded Koszul quasi-hereditary algebra is quasi-hereditary, then the algebra must be standard Koszul. This implies that the class of graded standard Koszul quasi-hereditary algebras is closed with respect to homological duality. Another immediate consequence is the fact that all algebras corresponding to the blocks of the category O are standard Koszul.

We study asymptotic behaviour of graded and non-graded codimensions of simple Jordan superalgebras over a field of characteristic zero. It is known that the PI-exponent of any finite-dimensional associative or Jordan or Lie algebra A is a non-negative integer less than or equal to the dimension of algebra A. Moreover, the PI-exponent is equal to the dimension if and only if A is simple provided that the base field is algebraically closed. In the present paper we prove that for a Jordan superalgebra P(t) = H(Mt∣t, trp) its non-graded and ℤ2-graded exponents are strictly less than dim P(t). In particular, exp P(2) is fractional.

The motivation of this paper is to study the natural quiver of an artinian algebra, a new kind of quivers, as a tool independing upon the associated basic algebra. In \cite{Li}, the notion of the natural quiver of an artinian algebra was introduced and then was used to generalize the Gabriel theorem for non-basic artinian algebras splitting over radicals and non-basic finite dimensional algebras with 2-nilpotent radicals via pseudo path algebras and generalized path algebras respectively. In this paper, firstly we consider the relationship between the natural quiver and the ordinary quiver of a finite dimensional algebra. Secondly, the generalized Gabriel theorem is obtained for radical-graded artinian algebras. Moreover, Gabriel-type algebras are introduced to outline those artinian algebras satisfying the generalized Gabriel theorem here and in \cite{Li}. For such algebras, the uniqueness of the related generalized path algebra and quiver holds up to isomorphism in the case when the ideal is admissible. For an artinian algebra, there are two basic algebras, the first is that associated to the algebra itself; the second is that associated to the correspondent generalized path algebra. In the final part, it is shown that for a Gabriel-type artinian algebra, the first basic algebra is a quotient of the second basic algebra. In the end, we give an example of a skew group algebra in which the relation between the natural quiver and the ordinary quiver is discussed., 14 pages

The purpose of this paper is to investigate general valuations and convex subrings of a commutative ring with higher level preordering. By defining the compatibility between valuations and preorderings (or orderings) of higher level in a commutative ring, we discuss the interplay between valuations and orderings of higher level. It is shown that the convex subrings are exactly valuation subrings compatible with some ordering. Moreover, the elements in the convex hull of a subring with respect to a given preordering are described. In this paper, the emphasis on Manis valuations is removed.

Foxby (Math Scand 2:175–186, 1971–1972) showed that a Cohen-Macaulay module over a Gorenstein local ring has finite projective dimension if and only if its canonical module has finite injective dimension. In this paper, we establish the result given by Foxby in a general setting. As a byproduct, some criteria to detect the Cohen-Macaulay property of a ring are provided in terms of intrinsic properties of certain local cohomology modules. Also, as an application, we show that any Cohen-Macaulay module that has a canonical module with finite injective dimension satisfies the Auslander–Reiten conjecture.

We compute the Nakayama automorphism of a PBW-deformation of a Koszul Artin-Schelter Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be Calabi-Yau. The relations between the Calabi-Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this paper is elementary and based on linear algebra. The results obtained in this paper will be applied in a subsequent paper., Comment: to appear at Manuscripta Math

The main purpose of this paper is to investigate the commuting conditions for prime MA-semirings through Jordan ideals and generalized derivations which are responsible to extend a few results of 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.

The only hitherto known Kurosh–Amitsur prime radical of near-rings is the equiprime radical which is also ideal-hereditary in the variety of zero-symmetric near-rings. In this paper an other prime radical is introduced in near-rings which is Kurosh-Amitsur but it fails to be ideal-hereditary.

A ring R is called right Ikeda–Nakayama ring (right IN-ring) if for any two right ideals I, J of R, $$l(I)+l(J)=l(I \cap J)$$ . In this paper, we introduce the concept of Essential Ikeda–Nakayama rings (EIN-rings) as a generalization of right IN-rings. This class of rings includes semiprime rings. We prove that for a left nonsingular EIN-ring R, closed ideals of R are right annihilator in R. We show that the class of EIN-rings is closed under direct product and upper triangular matrix rings. Furthermore, a ring R is an Armendariz EIN-ring if and only if R[x] is an EIN-ring.

Let $R$ be the power series ring or the polynomial ring over a field $k$ and let $I $ be an ideal of $R.$ Macaulay proved that the Artinian Gorenstein $k$-algebras $R/I$ are in one-to-one correspondence with the cyclic $R$-submodules of the divided power series ring $\Gamma. $ The result is effective in the sense that any polynomial of degree $s$ produces an Artinian Gorenstein $k$-algebra of socle degree $s.$ In a recent paper, the authors extended Macaulay's correspondence characterizing the $R$-submodules of $\Gamma $ in one-to-one correspondence with Gorenstein d-dimensional $k$-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein $k$-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the $G$-admissible submodules of $\Gamma. $ Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed., Comment: To appear in Trans. Am. Math. Soc

In this paper the center of a Leavitt path algebra is computed for a wide range of situations. A basis as a $K$-vector space is found for $Z(L(E))$ when $L(E)$ enjoys some finiteness condition such as being artinian, semisimple, noetherian and locally noetherian. The main result of the paper states that a simple Leavitt path algebra $L(E)$ is central (i.e. the center reduces to the base field $K$) when $L(E)$ is unital and has zero center otherwise. Finally, this result is extended, under some mild conditions, to the case of exchange Leavitt path algebras.

Every indecomposable module over a generalized uniserial ring is uniserial, hence local. This motivates one to study rings R satisfying the condition (*): R is a right artinian ring such that every finitely generated, indecomposable right R-module is local. The rings R satisfying (*) have been recently studied by Singh and Al-Bleahed (2004), they have proved some results giving the structure of local right R-modules. In this paper some more structure theorems for local right R-modules are proved. Examples given in this paper show that a rich class of rings satisfying condition (*) can be constructed. Using these results, it is proved that any ring R satisfying (*) is such that mod-R is of finite representation type. It follows from a theorem by Ringel and Tachikawa that any right R-module is a direct sum of local modules. If M is right module over a right artinian ring such that any finitely generated submodule of any homomorphic image of M is a direct sum of local modules, it is proved that it is a direct sum of local modules. This provides an alternative proof for that any right module over a right artinian ring R satisfying (*) is a direct sum of local modules.

In the present paper it is proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed *-graded exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary ℤ2-grading and graded involution.

Let R be a commutative ring with identity and M be an R-module. A non-zero submodule N of M is said to be a sum 2-irreducible submodule if whenever $$N=H_1+H_2+H_3$$ for submodules $$H_1$$ , $$H_2$$ and $$H_3$$ of M, then either $$N=H_1+H_2$$ or $$N=H_2+H_3$$ or $$N=H_1+H_3$$ . In this paper, we will introduce the concept of strongly sum 2-irreducible submodules of M as a generalization of sum 2-irreducible submodules of M and obtain some related results.

In this paper we introduce the concept of Rees closure for subalgebras of universal algebras. We denote by △𝐴 the identity relation on 𝐴. A subalgebra 𝐵 of algebra 𝐴 is called a Rees subalgebra whenever 𝐵2 ∪ △𝐴 is a congruence on 𝐴. A congruence 𝜃 of algebra 𝐴 is called a Rees congruence if 𝜃 = 𝐵2 ∪△𝐴 for some subalgebra 𝐵 of 𝐴. We define a Rees closure operator by mapping arbitrary subalgebra 𝐵 of algebra 𝐴 into the smallest Rees subalgebra that contains 𝐵. It is shown that in the general case the Rees closure does not commute with the operation ∧ on the lattice of subalgebras of universal algebra. Consequently, in the general case, a lattice of Rees subalgebras is not a sublattice of lattice of subalgebras. A non-one-element universal algebra 𝐴 is called a Rees simple algebra if any Rees congruence on 𝐴 is trivial. We characterize Rees simple algebras in terms of Rees closure. Universal algebra is called an algebra with operators if it has an additional set of unary operations acting as endomorphisms with respect to basic operations. We described Rees simple algebras in some subclasses of the class of algebras with one operator and a ternary basic operation. For algebras from these classes, the structure of lattice of Rees subalgebras is described. Necessary and sufficient conditions for the lattice of Rees subalgebras of algebras from these classes to be a chain are obtained.

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n\ge0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim(A_n)=r^nn!$ where $r = \dim(A_1)$. In the case $r=1$ we give a conjectural classification. We then investigate a quantum version of the main theorem. We conclude with some open problems and a categorification of the construction. This paper is a full version of the summary arXiv: 0710.3744., 18 pages. Minor changes, references updated

The main aim of the paper is to give a socle theory for Leavitt path algebras of arbitrary graphs. We use both the desingularization process and combinatorial methods to study Morita invariant properties concerning the socle and to characterize it, respectively. Leavitt path algebras with nonzero socle are described as those which have line points, and it is shown that the line points generate the socle of a Leavitt path algebra, extending so the results for row-finite graphs in the previous paper [12] (but with different methods). A concrete description of the socle of a Leavitt path algebra is obtained: it is a direct sum of matrix rings (of finite or infinite size) over the base field. New proofs of the Graded Uniqueness and of the Cuntz-Krieger Uniqueness Theorems are given, shorthening significantly the original ones., 20 pgs

In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander-Reiten quiver of the subcategories is obtained only by the Ausalander-Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, whenever $\Lambda$ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the $2 \times 2$ upper triangular matrix algebra over $\Lambda$ and the stable Auslander algebra of $\Lambda$ can be estimated by the category of modules over the stable Cohen-Macaulay Auslander algebra of $\Lambda$., Comment: Accepted for publication in Forum Mathematicum

A classical theorem of Zimmermann describes the relation between almost split sequences in the category of finitely presented modules and those in the category of all modules over some fixed ring. An analogue of Auslander–Reiten triangles in triangulated categories is proved in this paper. This is used to explain the relation between different existence results for Auslander–Reiten triangles, which are based either on Brown’s representability theorem, or on the existence of Serre functors. This paper is a continuation of [14], where Brown representability was introduced as a foundation for a general Auslander–Reiten theory in triangulated categories. More recently, Auslander–Reiten triangles have played an important role in work of Reiten and van den Bergh [18] and Jorgensen [11]. In [18], the existence of Auslander–Reiten triangles is translated into the existence of a Serre functor in the sense of Bondal and Kapranov [4]. This is used for a classification of the noetherian hereditary abelian categories satisfying Serre duality. In [11], the singular cochain differential graded algebra of a simply connected space is studied. The existence of

The paper establishes a relationship between finite separable extensions and norm groups of strictly quasilocal fields with Henselian discrete valuations, which yields a generally nonabelian one-dimensional local class field theory., Comment: 14 pages; revised form, to appear in manuscripta mathematica, paper [6] from the list of references has been published in the Proceedings of ICTAMI 05, Alba Iulia, Romania, 15.9-18.9. 2005; Acta Universitatis Apulensis 10/2005, 149-167

We consider the so-called Derksen group which is a subgroup of the polynomial automorphism group of the polynomial ring in n variables over a field. The Derksen group is generated by affine automorphisms and one particular non-linear automorphism. Derksen (1994) proved that if the characteristic of the underlying field is zero and n ≥ 3, then the Derksen group is equal to the entire tame subgroup. The result is called Derksen’s Theorem. It is quite natural to ask whether the same property holds for positive characteristic. In this paper, we point out that the question can be easily answered negatively when the underlying field is of characteristic two. We shall also prove that the permutation group induced by the Derksen group over a finite field of characteristic two is a proper subgroup of the alternating group on the n-dimensional linear space over the finite field. This is a stronger result that Derksen’s Theorem does not hold when the underlying field is a finite field of characteristic two.

J. Zelmanowitz introduced the concept of a ring, which we call a zip ring. In this paper we characterize a commutative Bezout domain whose finite homomorphic images are zip rings modulo its nilradical.

In this paper, we obtain some relations between Noetherian and Artinian BL-algebras. Further, we derive some theorems and lemmas for composition series, Artinian BL-argebera, as well as the relations between Noetherian, Artinian and short exact sequences. We further study the Noetherian and Artinian over homomorphism BL-algebras and obtain some new results concerning to the essential deductive systems of BL-algebras.

In this paper, we investigate the linkage of ideals, in Noetherian and non-Noetherian settings, with the aim to establish new characterizations of classical notions of domains through linkage theory. Two main results assert that a Noetherian domain is Dedekind if and only if it has the primary linkage property; and a domain is almost Dedekind (resp., Prufer) if and only if it has the linkage (resp., finite linkage) property. Also, we prove that a finite-dimensional valuation domain is a DVR (i.e., Noetherian) if and only if it has the primary linkage property.

It is well known that the decomposition of injective modules over Noether rings and the decomposition of projective modules over Artinian rings are some of the most beautiful and important results in commutative algebra. The goal of the paper is to prove similar results for graded rings; this is important to understand the structure of modules over graded rings. The obtained results include structural theorems for gr-injective modules over gr-Noetherian G-graded commutative rings and for gr-finitely generated gr-projective modules over gr-Artinian G-graded commutative rings. Bibliography: 13 titles.

The classical Gelfand–Kirillov dimension for algebras over fields has been extended recently by J. Bell and J.J Zhang to algebras over commutative domains. However, the behavior of this new notion has not been enough investigated for the principal algebraic constructions as polynomial rings, matrix rings, localizations, filtered–graded rings, skew PBW extensions, etc. In this paper, we present complete proofs of the computation of this more general dimension for the mentioned algebraic constructions for algebras over commutative domains. The Gelfand–Kirillov dimension for modules and the Gelfand–Kirillov transcendence degree will be also considered. The obtained results can be applied in particular to algebras over the ring of integers, i.e, to arbitrary rings.

Let R be a 2-torsion free prime ring with center Z(R) and a nonzero Jordan ideal J. In this paper, our main objective is to prove that: If $$F:R\rightarrow R$$ is a generalized derivation associated with a derivation d of R such that $$[F(u),u]_{k}\in Z(R)$$ for all $$u\in J$$ and a fixed integer $$k\ge 1,$$ then either R is commutative or there exists some $$a\in C$$ such that $$F(x)=ax$$ for all $$x\in R,$$ which extends a result of Soufi and Aboubakr (Turk. J. Math. 38:233–239, 2014, Theorem 3.2). Moreover, a Jordan ideal which is also a subring seems close to an ideal. Therefore, results on Jordan ideals (which are not necessarily subrings) are more interesting. We show that all the theorems of Soufi and Aboubakr [20] are true for Jordan ideals (which are not necessarily subrings).

Relative Auslander algebras were introduced and studied by Beligiannis. In this paper, we apply intermediate extension functors associated to certain recollements of functor categories to study them. In particular, we study the existence of tilting-cotilting modules over such algebras. As a consequence, it will be shown that two Gorenstein algebras of G-dimension 1 being of finite Cohen-Macaulay type are Morita equivalent if and only if their Cohen-Macaulay Auslander algebras are Morita equivalent.