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Let Λ be an artin algebra and C be a functorially finite subcategory of mod Λ which contains Λ or D Λ. 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. [ABSTRACT FROM AUTHOR]

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 an essentially 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, Ind (M) as a subcategory of Mod M Eff (M) falls short of being a d -cluster tilting subcategory since it satisfies all properties of a d -cluster tilting subcategory except d -rigidity. 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 Iyama's question about the finiteness of M. Furthermore, for general d , we address the problem and present several equivalent conditions for Iyama's question. [ABSTRACT FROM AUTHOR]

We study Morita equivalence and Morita duality for rings with local units. We extend Auslander’s results on the theory of Morita equivalence and the Azumaya–Morita duality theorem to rings with local units. As a consequence, we give a version of Morita theorem and Azumaya–Morita duality theorem over rings with local units in terms of their full subcategory of finitely generated projective unitary modules and full subcategory of finitely generated injective unitary modules. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Let |$\mathcal{M}$| be a small |$n$| -abelian category. We show that the category of finitely presented functors |${\operatorname{mod}}$| - |$\mathcal{M}$| modulo the subcategory of effaceable functors |${\operatorname{mod}}_0$| - |$\mathcal{M}$| has an |$n$| -cluster tilting subcategory, which is equivalent to |$\mathcal{M}$|⁠. This gives a higher-dimensional version of Auslander's formula. [ABSTRACT FROM AUTHOR]

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 → A b has finite length. As a consequence we show that if mod Λ has an 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. [ABSTRACT FROM AUTHOR]

In this paper, we study the category of finitely generated modules over a class of right 4 -Nakayama artin algebras. This class of algebras appear naturally in the study of representation-finite artin algebras. First, we give a characterization of right 4 -Nakayama artin algebras. Then, we classify finitely generated indecomposable right modules over right 4 -Nakayama artin algebras. We also compute almost split sequences for the class of right 4 -Nakayama artin algebras. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that a directed graph $E$ is a finite graph with no sinks if and only if, for each commutative unital ring $R$ , the Leavitt path algebra $L_{R}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the $C^{\ast }$ -algebra $C^{\ast }(E)$ is unital and $\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. Let $k$ be a field and $k^{\times }$ be the group of units of $k$. When $\text{rank}(k^{\times }) , we show that the Leavitt path algebra $L_{k}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if $L_{k}(E)$ is unital and $\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$. We also show that any unital $k$ -algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras. [ABSTRACT FROM AUTHOR]

From the viewpoint of higher homological algebra, we introduce pure semisimple n -abelian categories, which are analogs of pure semisimple abelian categories. Let Λ be an Artin algebra and M be an n -cluster tilting subcategory of Mod-Λ. We show that M is pure semisimple if and only if each module in M is a direct sum of finitely generated modules. Let m be an n -cluster tilting subcategory of mod-Λ. We show that Add (m) is an n -cluster tilting subcategory of Mod-Λ if and only if m has an additive generator if and only if Mod (m) is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras. [ABSTRACT FROM AUTHOR]

Let Λ be a cluster-tilted algebra of finite type over an algebraically closed field and let B be one of the associated tilted algebras. We show that the B-modules, ordered from 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. [ABSTRACT FROM AUTHOR]

Abstract 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 k R . 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. [ABSTRACT FROM AUTHOR]

Abstract For any unital commutative ring R and for any graph E , we identify the commutative core of the Leavitt path algebra of E with coefficients in R , which is a maximal commutative subalgebra of the Leavitt path algebra. Furthermore, we are able to characterize injectivity of representations which gives a generalization of the Cuntz–Krieger uniqueness theorem. [ABSTRACT FROM AUTHOR]

For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e., gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a clustertilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons’s conjecture. [ABSTRACT FROM AUTHOR]

For any field K and for a completely arbitrary graph E, we characterize the Leavitt path algebras L K( E) that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it actually does so as a direct sum of Leavitt path algebras for some suitable graphs. Under certain finiteness conditions, a unique indecomposable decomposition exists. [ABSTRACT FROM AUTHOR]

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