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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]

In this paper, we introduce |$n\mathbb{Z}$| -abelian and |$n\mathbb{Z}$| -exact categories by axiomatizing properties of |$n\mathbb{Z}$| -cluster tilting subcategories. We study these categories and show that every |$n\mathbb{Z}$| -cluster tilting subcategory of an abelian (resp. exact) category has a natural structure of an |$n\mathbb{Z}$| -abelian (resp. |$n\mathbb{Z}$| -exact) category. Also, we show that every small |$n\mathbb{Z}$| -abelian category arises in this way, and discuss the problem for |$n\mathbb{Z}$| -exact categories. [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]

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]

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. [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]

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]

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]

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]

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]