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In this paper we introduce $n\mathbb{Z}$-abelian and $n\mathbb{Z}$-exact categories by axiomatising properties of $n\mathbb{Z}$-cluster tilting subcategories. We study this 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 arise in this way, and discuss the problem for $n\mathbb{Z}$-exact categories.

Let $\mathcal{M}$ be a small category. The completion of $\mathcal{M}$ denoted by $Ind(\mathcal{M})$, is by definition the universal completion of $\mathcal{M}$ with respect to filtered colimits. In this paper we study $Ind(\mathcal{M})$, for a small $d$-abelian category $\mathcal{M}$. We show that $Ind(\mathcal{M})$ is equivalent to the full subcategory $\mathcal{L}_d(\mathcal{M})$ of $Mod \mathcal{M}$, consist of left $d$-exact functors. We also show that $Ind(\mathcal{M})$ as a subcategory of $\frac{Mod \mathcal{M}}{Eff(\mathcal{M})}$, is too close to be a $d$-cluster tilting subcategory and conjecture that it is indeed a $d$-cluster tilting subcategory.

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.

A ring $��$ is called right K��the if every right $��$-module is a direct sum of cyclic modules. In this paper, we give a characterization of basic hereditary right K��the rings in terms of their Coxeter valued quivers. Also we give a characterization of basic right K��the rings with radical square zero. Therefore we give a solution of the K��the's problem in this cases., to appear in Canad. J. Math

We show that $E$ is a finite graph with no sinks if and only if the Leavitt path algebra $L_R(E)$ is isomorphic to an algebraic Cuntz-Krieger algebra if and only if the $C^*$-algebra $C^*(E)$ is unital and $rank(K_0(C^*(E)))=rank(K_1(C^*(E)))$. When $k$ is a field and $rank(k^{\times})< \infty$, 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 $rank(K_1(L_k(E)))=(rank(k^{\times})+1)rank(K_0(L_k(E)))$. 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., to appear in J. Australian Math. Soc

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 K��the rings, which is a generalization of K��the-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 K��the ring if and only if $R$ is an artinian left multiplicity-free top ring.