Alireza Nasr-Isfahani - Academia.edu (original) (raw)

Papers by Alireza Nasr-Isfahani

Some relationships between representations of a hypergroup X, its algebras, and positive definite... more Some relationships between representations of a hypergroup X, its algebras, and positive definite functions on X are studied. Also, various types of convergence of positive definite functions on X are discussed.

arXiv (Cornell University), Apr 15, 2018

Let Λ be a cluster-tilted algebra of finite type over an algebraically closed field and B be one ... more Let Λ be a cluster-tilted algebra of finite type over an algebraically closed field and B be one of the associated tilted algebras. We show that the B-modules, ordered form right to left in the Auslander-Reiten quiver of Λ form a maximal forward hom-orthogonal sequence of Λ-modules whose dimension vectors form the c-vectors of a maximal green sequence for Λ. Thus we give a proof of Igusa-Todorov's conjecture.

arXiv (Cornell University), Oct 1, 2022

Let A be a finite-dimensional algebra, and M be a d-cluster tilting subcategory of mod A. From th... more Let A be a finite-dimensional algebra, and M be a d-cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d-cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as a small d-abelian category, known to be equivalent to a d-cluster tilting subcategory of an abelian category A. The completion of M, denoted by Ind(M), is defined as the universal completion of M with respect to filtered colimits. We explore Ind(M) and demonstrate its equivalence to the full subcategory L d (M) of Mod M, comprising left d-exact functors. Notably, while Ind(M) as a subcategory of Mod M Eff(M) , satisfies all properties of a d-cluster tilting subcategory except d-rigidity, it falls short of being a d-cluster tilting category. For a d-cluster tilting subcategory M of mod A, − → M, consists of all filtered colimits of objects from M, is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M is a d-rigid subcategory remains unanswered. However, if it is indeed d-rigid, it qualifies as a d-cluster tilting subcategory. In the case d = 2, employing cotorsion theory, we establish that − → M is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether − → M is a d-cluster tilting subcategory of Mod A appears to be equivalent to the Iyama's qestion about the finiteness of M. Furthermore, for general d, we address the problem and present several equivalent conditions for the Iyama's question.

Let R be a ring, α an automorphism, and δ an α-derivation of R. If the classical quotient ring Q ... more Let R be a ring, α an automorphism, and δ an α-derivation of R. If the classical quotient ring Q of R exists, then R is weak α-skew Armendariz if and only if Q is weak α-skew Armendariz.

Proceedings of the American Mathematical Society, Nov 21, 2016

Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalg... more Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra M inside the Kumjian-Pask algebra KP R (Λ). We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of KP R (Λ) is injective if and only if it is injective on M.

arXiv (Cornell University), Nov 7, 2020

We introduce the notion of a lower bound cluster algebra generated by projective cluster variable... more We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case we use our results to construct a basis for the cluster algebra. We also show that any coefficient-free cluster algebra of types A n or A n is equal to the corresponding lower bound cluster algebra generated by projective cluster variables.

arXiv (Cornell University), Dec 9, 2017

A ring R is called left k-cyclic if every left R-module is a direct sum of indecomposable modules... more A ring R is called left k-cyclic if every left R-module is a direct sum of indecomposable modules which are homomorphic image of R R k. In this paper, we give a characterization of left k-cyclic rings. As a consequence, we give a characterization of left Köthe rings, which is a generalization of Köthe-Cohen-Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left k-cyclic ring. As a corollary, we show that R is Morita equivalent to a basic left Köthe ring if and only if R is an artinian left multiplicity-free top ring.

arXiv (Cornell University), Jun 21, 2021

Let Λ be an artin algebra and M be an n-cluster tilting subcategory of mod Λ. We show that M has ... more Let Λ be an artin algebra and M be an n-cluster tilting subcategory of mod Λ. We show that M has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of M if and only if every effaceable functor M → Ab has finite length. As a consequence we show that if mod Λ has n-cluster tilting subcategory of finite type then the n-almost split sequences form a basis for the relations for the Grothendieck group of Λ.

arXiv (Cornell University), Jun 10, 2020

Let M be a small n-abelian category. We show that the category of finitely presented functors mod... more Let M be a small n-abelian category. We show that the category of finitely presented functors mod-M modulo the subcategory of effaceable functors mod 0-M has an n-cluster tilting subcategory which is equivalent to M. This gives a higherdimensional version of Auslander's formula.

arXiv (Cornell University), Jul 9, 2022

Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or... more Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or DΛ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if rad ∞ C vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C-approximations and right C-approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C.

Journal of Algebra, Dec 1, 2020

We prove a higher-dimensional version of the Freyd-Mitchell embedding theorem for n-abelian categ... more We prove a higher-dimensional version of the Freyd-Mitchell embedding theorem for n-abelian categories. More precisely, for a positive integer n and a small n-abelian category M, we show that M is equivalent to a full subcategory of an abelian category L 2 (M, G), where L 2 (M, G) is the category of absolutely pure group valued functors over M. We also show that n-kernels and n-cokernels in M are precisely exact sequences of L 2 (M, G) with terms in M.

arXiv (Cornell University), Aug 31, 2021

Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified ve... more Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified version of higher Auslander correspondence in the context of exact categories. We define n-Auslander exact categories and show that there is a bijection between the equivalence classes of n-cluster tilting subcategories of exact categories and the equivalence classes of n-Auslander exact categories.

arXiv (Cornell University), May 11, 2018

In this paper we study the category of finitely generated modules over a right 3-Nakayama artin a... more In this paper we study the category of finitely generated modules over a right 3-Nakayama artin algebra. First we give a characterization of right 3-Nakayama artin algebras and then we give a complete list of non-isomorphic finitely generated indecomposable modules over any right 3-Nakayama artin algebra. Also we compute all almost split sequences for the class of right 3-Nakayama artin algebras. Finally, we classify finite dimensional right 3-Nakayama algebras in terms of their quivers with relations. 1. introduction Let R be a commutative artinian ring. An R-algebra Λ is called an artin algebra if Λ is finitely generated as a R-module. Let Λ be an artin algebra. A right Λ-module M is called uniserial (1-factor serial) if it has a unique composition series. An artin algebra Λ is called Nakayama algebra if any indecomposable right Λ-module is uniserial. The class of Nakayama algebras is one the important class of representation finite algebras whose representation theory completely understood [3]. According to [5, Definition 2.1], a non-uniserial right Λ-module M of length l is called n-factor serial (l ≥ n > 1), if M rad l−n (M) is uniserial and M rad l−n+1 (M) is not uniserial. An artin algebra Λ is called right n-Nakayama if every indecomposable right Λ-module is i-factor serial for some 1 ≤ i ≤ n and there exists at least one indecomposable n-factor serial right Λ-module [5, Definition 2.2]. The authors in [5] showed that the class of right n-Nakayama algebras provide a nice partition of the class of representation finite artin algebras. More precisely, the authors proved that an artin algebra Λ is representation finite if and only if Λ is right n-Nakayama for some positive integer n [5, Theorem 2.18]. The first part of this partition is the class of Nakayama algebras and the second part is the class of right 2-Nakayama algebras. Indecomposable modules and almost split sequences for the class of right 2-Nakayama algebras are classified in section 5 of [5]. In this paper we will study the class of right 3-Nakayama algebras. We first show that an artin algebra Λ which is neither Nakayama nor right 2-Nakayama is right 3-Nakayama if and only if every indecomposable right Λ-module of length greater than 4 is uniserial and every indecomposable right Λmodule of length 4 is local. Then we classify all indecomposable modules and almost split sequences over a right 3-Nakayama artin algebra. We also show that finite dimensional right 3-Nakayama algebras are special biserial and we describe all finite dimensional right 3-Nakayama algebras by their quivers and relations. Riedtmann in [6] and [7], by using the covering theory, classified representation-finite self-injective algebras.

Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified ve... more Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified version of higher Auslander correspondence in the context of exact categories. We define n-Auslander exact categories and show that there is a bijection between the equivalence classes of n-cluster tilting subcategories of exact categories and the equivalence classes of n-Auslander exact categories.

arXiv: Representation Theory, 2018

We characterize right 444-Nakayama artin algebras which appear naturally in the study of represen... more We characterize right 444-Nakayama artin algebras which appear naturally in the study of representation-finite artin algebras. For a right 444-Nakayama artin algebra Lambda\LambdaLambda, we classify all finitely generated indecomposable right Lambda\LambdaLambda-modules and then we compute all almost split sequences over Lambda\LambdaLambda. We also give a characterization of right 444-Nakayama finite dimensional KKK-algebras in terms of their quivers with relations.

Journal of Algebra, 2020

Let M be a small n-abelian category. We show that the category of absolutely pure group valued fu... more Let M be a small n-abelian category. We show that the category of absolutely pure group valued functors over M, denote by L 2 (M, G), is an abelian category and M is equivalent to a full subcategory of L 2 (M, G) in such a way that n-kernels and n-cokernels are precisely exact sequences of L 2 (M, G) with terms in M. This gives a higher-dimensional version of the Freyd-Mitchell embedding theorem for n-abelian categories.

Algebras and Representation Theory, 2019

In this paper we study right n-Nakayama algebras. Right n-Nakayama algebras appear naturally in t... more In this paper we study right n-Nakayama algebras. Right n-Nakayama algebras appear naturally in the study of representation-finite algebras. We show that an artin algebra Λ is representation-finite if and only if Λ is right n-Nakayama for some positive integer n. We classify hereditary right n-Nakayama algebras. We also define right n-coNakayama algebras and show that an artin algebra Λ is right n-coNakayama if and only if Λ is left n-Nakayama. We then study right 2-Nakayama algebras. We show how to compute all the indecomposable modules and almost split sequences over a right 2-Nakayama algebra. We end by classifying finite dimensional right 2-Nakayama algebras in terms of their quivers with relations. 1. Introduction Let R be a commutative artinian ring. An R-algebra Λ is called an artin algebra if Λ is finitely generated as an R-module. Given an artin algebra Λ, it is a quite natural question to ask for the classification of all the indecomposable finitely generated right Λ-modules. Only for few classes of algebras such a classification is known, one of the first such class were the Nakayama algebras. A Nakayama algebra Λ is an algebra such that the indecomposable projective right Λ-modules as well as the indecomposable injective right Λ-modules are uniserial. This then implies that all the indecomposable right Λmodules are uniserial. Nakayama algebras were studied by Tadasi Nakayama who called them generalized uniserial rings [9, 10]. A right Λ-module M is called uniserial if it has a unique composition series. Uniserial modules are the simplest indecomposables and this makes it interesting to understand their role in the category mod(Λ) of finitely generated right Λ-modules. An artin algebra Λ is said to be representation-finite, provided there are only finitely many isomorphism classes of indecomposable right Λ-modules. In representation theory, representation-finite algebras are of particular importance since in this case one has a complete combinatorial description of the module category in terms of the Auslander-Reiten quiver. The class of Nakayama algebras is one of the fundamental classes of representation-finite algebras whose representation theory is completely understood. In this paper we introduce the notion of n-factor serial modules. We say that a nonuniserial right Λ-module M of length l is n-factor serial (l ≥ n > 1), if M rad l−n (M) is uniserial and M rad l−n+1 (M) is not uniserial. In some sense, n is an invariant that measures how far M is from being uniserial. We say that an artin algebra Λ is right n-Nakayama if every finitely generated indecomposable right Λ-module is i-factor serial for some 1 i n

Communications in Algebra, 2016

ABSTRACT In this article, we show that there exists an SCN ring R such that the polynomial ring R... more ABSTRACT In this article, we show that there exists an SCN ring R such that the polynomial ring R[x] is not SCN. This answers a question posed by T. K. Kwak et al. in [2].

Communications in Algebra, 2016

Let R be a ring with an automorphism α and a derivation δ. In this article we provide necessary a... more Let R be a ring with an automorphism α and a derivation δ. In this article we provide necessary and su cient conditions for a skew polynomial ring R[x; α] and di erential polynomial ring R[x; δ] to be 2-primal. We compute the Jacobson radical and the set of unit elements of a 2-primal skew polynomial ring R[x; α] and di erential polynomial ring R[x; δ]. Also we establish the lower bounds on the stable range of a 2-primal skew polynomial ring R[x; α] and di erential polynomial ring R[x; δ]. As an application we show that if R is 2-primal then the nth Weyl algebra over R is 2-primal and in this case J(A n (R)) = A n (Niℓ * (R)). As a consequence, we extend and unify several known results of [4], [8], [10], [18], [19], and [22].

Proceedings of the American Mathematical Society, 2016

Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalg... more Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra M inside the Kumjian-Pask algebra KP R (Λ). We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of KP R (Λ) is injective if and only if it is injective on M.

Some relationships between representations of a hypergroup X, its algebras, and positive definite... more Some relationships between representations of a hypergroup X, its algebras, and positive definite functions on X are studied. Also, various types of convergence of positive definite functions on X are discussed.

arXiv (Cornell University), Apr 15, 2018

Let Λ be a cluster-tilted algebra of finite type over an algebraically closed field and B be one ... more Let Λ be a cluster-tilted algebra of finite type over an algebraically closed field and B be one of the associated tilted algebras. We show that the B-modules, ordered form right to left in the Auslander-Reiten quiver of Λ form a maximal forward hom-orthogonal sequence of Λ-modules whose dimension vectors form the c-vectors of a maximal green sequence for Λ. Thus we give a proof of Igusa-Todorov's conjecture.

arXiv (Cornell University), Oct 1, 2022

Let A be a finite-dimensional algebra, and M be a d-cluster tilting subcategory of mod A. From th... more Let A be a finite-dimensional algebra, and M be a d-cluster tilting subcategory of mod A. From the viewpoint of higher homological algebra, a natural question to ask is when M induces a d-cluster tilting subcategory in Mod A. In this paper, we investigate this question in a more general form. We consider M as a small d-abelian category, known to be equivalent to a d-cluster tilting subcategory of an abelian category A. The completion of M, denoted by Ind(M), is defined as the universal completion of M with respect to filtered colimits. We explore Ind(M) and demonstrate its equivalence to the full subcategory L d (M) of Mod M, comprising left d-exact functors. Notably, while Ind(M) as a subcategory of Mod M Eff(M) , satisfies all properties of a d-cluster tilting subcategory except d-rigidity, it falls short of being a d-cluster tilting category. For a d-cluster tilting subcategory M of mod A, − → M, consists of all filtered colimits of objects from M, is a generating-cogenerating, functorially finite subcategory of Mod A. The question of whether M is a d-rigid subcategory remains unanswered. However, if it is indeed d-rigid, it qualifies as a d-cluster tilting subcategory. In the case d = 2, employing cotorsion theory, we establish that − → M is a 2-cluster tilting subcategory if and only if M is of finite type. Thus, the question regarding whether − → M is a d-cluster tilting subcategory of Mod A appears to be equivalent to the Iyama's qestion about the finiteness of M. Furthermore, for general d, we address the problem and present several equivalent conditions for the Iyama's question.

Let R be a ring, α an automorphism, and δ an α-derivation of R. If the classical quotient ring Q ... more Let R be a ring, α an automorphism, and δ an α-derivation of R. If the classical quotient ring Q of R exists, then R is weak α-skew Armendariz if and only if Q is weak α-skew Armendariz.

Proceedings of the American Mathematical Society, Nov 21, 2016

Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalg... more Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra M inside the Kumjian-Pask algebra KP R (Λ). We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of KP R (Λ) is injective if and only if it is injective on M.

arXiv (Cornell University), Nov 7, 2020

We introduce the notion of a lower bound cluster algebra generated by projective cluster variable... more We introduce the notion of a lower bound cluster algebra generated by projective cluster variables as a polynomial ring over the initial cluster variables and the so-called projective cluster variables. We show that under an acyclicity assumption, the cluster algebra and the lower bound cluster algebra generated by projective cluster variables coincide. In this case we use our results to construct a basis for the cluster algebra. We also show that any coefficient-free cluster algebra of types A n or A n is equal to the corresponding lower bound cluster algebra generated by projective cluster variables.

arXiv (Cornell University), Dec 9, 2017

A ring R is called left k-cyclic if every left R-module is a direct sum of indecomposable modules... more A ring R is called left k-cyclic if every left R-module is a direct sum of indecomposable modules which are homomorphic image of R R k. In this paper, we give a characterization of left k-cyclic rings. As a consequence, we give a characterization of left Köthe rings, which is a generalization of Köthe-Cohen-Kaplansky theorem. We also characterize rings which are Morita equivalent to a basic left k-cyclic ring. As a corollary, we show that R is Morita equivalent to a basic left Köthe ring if and only if R is an artinian left multiplicity-free top ring.

arXiv (Cornell University), Jun 21, 2021

Let Λ be an artin algebra and M be an n-cluster tilting subcategory of mod Λ. We show that M has ... more Let Λ be an artin algebra and M be an n-cluster tilting subcategory of mod Λ. We show that M has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of M if and only if every effaceable functor M → Ab has finite length. As a consequence we show that if mod Λ has n-cluster tilting subcategory of finite type then the n-almost split sequences form a basis for the relations for the Grothendieck group of Λ.

arXiv (Cornell University), Jun 10, 2020

Let M be a small n-abelian category. We show that the category of finitely presented functors mod... more Let M be a small n-abelian category. We show that the category of finitely presented functors mod-M modulo the subcategory of effaceable functors mod 0-M has an n-cluster tilting subcategory which is equivalent to M. This gives a higherdimensional version of Auslander's formula.

arXiv (Cornell University), Jul 9, 2022

Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or... more Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or DΛ. We use the concept of the infinite radical of C and show that C has an additive generator if and only if rad ∞ C vanishes. In this case we describe the morphisms in powers of the radical of C in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that C is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in C is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left C-approximations and right C-approximations of simple Λ-modules, we give other criteria to describe whether C is of finite representation type. In addition, we give a nilpotency index of the radical of C which is independent from the maximal length of indecomposable Λ-modules in C.

Journal of Algebra, Dec 1, 2020

We prove a higher-dimensional version of the Freyd-Mitchell embedding theorem for n-abelian categ... more We prove a higher-dimensional version of the Freyd-Mitchell embedding theorem for n-abelian categories. More precisely, for a positive integer n and a small n-abelian category M, we show that M is equivalent to a full subcategory of an abelian category L 2 (M, G), where L 2 (M, G) is the category of absolutely pure group valued functors over M. We also show that n-kernels and n-cokernels in M are precisely exact sequences of L 2 (M, G) with terms in M.

arXiv (Cornell University), Aug 31, 2021

Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified ve... more Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified version of higher Auslander correspondence in the context of exact categories. We define n-Auslander exact categories and show that there is a bijection between the equivalence classes of n-cluster tilting subcategories of exact categories and the equivalence classes of n-Auslander exact categories.

arXiv (Cornell University), May 11, 2018

In this paper we study the category of finitely generated modules over a right 3-Nakayama artin a... more In this paper we study the category of finitely generated modules over a right 3-Nakayama artin algebra. First we give a characterization of right 3-Nakayama artin algebras and then we give a complete list of non-isomorphic finitely generated indecomposable modules over any right 3-Nakayama artin algebra. Also we compute all almost split sequences for the class of right 3-Nakayama artin algebras. Finally, we classify finite dimensional right 3-Nakayama algebras in terms of their quivers with relations. 1. introduction Let R be a commutative artinian ring. An R-algebra Λ is called an artin algebra if Λ is finitely generated as a R-module. Let Λ be an artin algebra. A right Λ-module M is called uniserial (1-factor serial) if it has a unique composition series. An artin algebra Λ is called Nakayama algebra if any indecomposable right Λ-module is uniserial. The class of Nakayama algebras is one the important class of representation finite algebras whose representation theory completely understood [3]. According to [5, Definition 2.1], a non-uniserial right Λ-module M of length l is called n-factor serial (l ≥ n > 1), if M rad l−n (M) is uniserial and M rad l−n+1 (M) is not uniserial. An artin algebra Λ is called right n-Nakayama if every indecomposable right Λ-module is i-factor serial for some 1 ≤ i ≤ n and there exists at least one indecomposable n-factor serial right Λ-module [5, Definition 2.2]. The authors in [5] showed that the class of right n-Nakayama algebras provide a nice partition of the class of representation finite artin algebras. More precisely, the authors proved that an artin algebra Λ is representation finite if and only if Λ is right n-Nakayama for some positive integer n [5, Theorem 2.18]. The first part of this partition is the class of Nakayama algebras and the second part is the class of right 2-Nakayama algebras. Indecomposable modules and almost split sequences for the class of right 2-Nakayama algebras are classified in section 5 of [5]. In this paper we will study the class of right 3-Nakayama algebras. We first show that an artin algebra Λ which is neither Nakayama nor right 2-Nakayama is right 3-Nakayama if and only if every indecomposable right Λ-module of length greater than 4 is uniserial and every indecomposable right Λmodule of length 4 is local. Then we classify all indecomposable modules and almost split sequences over a right 3-Nakayama artin algebra. We also show that finite dimensional right 3-Nakayama algebras are special biserial and we describe all finite dimensional right 3-Nakayama algebras by their quivers and relations. Riedtmann in [6] and [7], by using the covering theory, classified representation-finite self-injective algebras.

Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified ve... more Inspired by the recent work of Henrard, Kvamme and van Roosmalen [17], we prove a categorified version of higher Auslander correspondence in the context of exact categories. We define n-Auslander exact categories and show that there is a bijection between the equivalence classes of n-cluster tilting subcategories of exact categories and the equivalence classes of n-Auslander exact categories.

arXiv: Representation Theory, 2018

We characterize right 444-Nakayama artin algebras which appear naturally in the study of represen... more We characterize right 444-Nakayama artin algebras which appear naturally in the study of representation-finite artin algebras. For a right 444-Nakayama artin algebra Lambda\LambdaLambda, we classify all finitely generated indecomposable right Lambda\LambdaLambda-modules and then we compute all almost split sequences over Lambda\LambdaLambda. We also give a characterization of right 444-Nakayama finite dimensional KKK-algebras in terms of their quivers with relations.

Journal of Algebra, 2020

Let M be a small n-abelian category. We show that the category of absolutely pure group valued fu... more Let M be a small n-abelian category. We show that the category of absolutely pure group valued functors over M, denote by L 2 (M, G), is an abelian category and M is equivalent to a full subcategory of L 2 (M, G) in such a way that n-kernels and n-cokernels are precisely exact sequences of L 2 (M, G) with terms in M. This gives a higher-dimensional version of the Freyd-Mitchell embedding theorem for n-abelian categories.

Algebras and Representation Theory, 2019

In this paper we study right n-Nakayama algebras. Right n-Nakayama algebras appear naturally in t... more In this paper we study right n-Nakayama algebras. Right n-Nakayama algebras appear naturally in the study of representation-finite algebras. We show that an artin algebra Λ is representation-finite if and only if Λ is right n-Nakayama for some positive integer n. We classify hereditary right n-Nakayama algebras. We also define right n-coNakayama algebras and show that an artin algebra Λ is right n-coNakayama if and only if Λ is left n-Nakayama. We then study right 2-Nakayama algebras. We show how to compute all the indecomposable modules and almost split sequences over a right 2-Nakayama algebra. We end by classifying finite dimensional right 2-Nakayama algebras in terms of their quivers with relations. 1. Introduction Let R be a commutative artinian ring. An R-algebra Λ is called an artin algebra if Λ is finitely generated as an R-module. Given an artin algebra Λ, it is a quite natural question to ask for the classification of all the indecomposable finitely generated right Λ-modules. Only for few classes of algebras such a classification is known, one of the first such class were the Nakayama algebras. A Nakayama algebra Λ is an algebra such that the indecomposable projective right Λ-modules as well as the indecomposable injective right Λ-modules are uniserial. This then implies that all the indecomposable right Λmodules are uniserial. Nakayama algebras were studied by Tadasi Nakayama who called them generalized uniserial rings [9, 10]. A right Λ-module M is called uniserial if it has a unique composition series. Uniserial modules are the simplest indecomposables and this makes it interesting to understand their role in the category mod(Λ) of finitely generated right Λ-modules. An artin algebra Λ is said to be representation-finite, provided there are only finitely many isomorphism classes of indecomposable right Λ-modules. In representation theory, representation-finite algebras are of particular importance since in this case one has a complete combinatorial description of the module category in terms of the Auslander-Reiten quiver. The class of Nakayama algebras is one of the fundamental classes of representation-finite algebras whose representation theory is completely understood. In this paper we introduce the notion of n-factor serial modules. We say that a nonuniserial right Λ-module M of length l is n-factor serial (l ≥ n > 1), if M rad l−n (M) is uniserial and M rad l−n+1 (M) is not uniserial. In some sense, n is an invariant that measures how far M is from being uniserial. We say that an artin algebra Λ is right n-Nakayama if every finitely generated indecomposable right Λ-module is i-factor serial for some 1 i n

Communications in Algebra, 2016

ABSTRACT In this article, we show that there exists an SCN ring R such that the polynomial ring R... more ABSTRACT In this article, we show that there exists an SCN ring R such that the polynomial ring R[x] is not SCN. This answers a question posed by T. K. Kwak et al. in [2].

Communications in Algebra, 2016

Let R be a ring with an automorphism α and a derivation δ. In this article we provide necessary a... more Let R be a ring with an automorphism α and a derivation δ. In this article we provide necessary and su cient conditions for a skew polynomial ring R[x; α] and di erential polynomial ring R[x; δ] to be 2-primal. We compute the Jacobson radical and the set of unit elements of a 2-primal skew polynomial ring R[x; α] and di erential polynomial ring R[x; δ]. Also we establish the lower bounds on the stable range of a 2-primal skew polynomial ring R[x; α] and di erential polynomial ring R[x; δ]. As an application we show that if R is 2-primal then the nth Weyl algebra over R is 2-primal and in this case J(A n (R)) = A n (Niℓ * (R)). As a consequence, we extend and unify several known results of [4], [8], [10], [18], [19], and [22].

Proceedings of the American Mathematical Society, 2016

Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalg... more Let Λ be a row-finite higher-rank graph with no sources. We identify a maximal commutative subalgebra M inside the Kumjian-Pask algebra KP R (Λ). We also prove a generalized Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras which says that a representation of KP R (Λ) is injective if and only if it is injective on M.