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Let $\Lambda$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of $\Lambda$-mod with $n\ge 2$. From the viewpoint of higher homological algebra, a question that naturally arose in [17] is when $\mathcal{M}$ induces an n-cluster tilting subcategory of $\Lambda$-Mod. In this paper, we answer this question and explore its connection to Iyama's question on the finiteness of n-cluster tilting subcategories of $\Lambda$-mod. In fact, our theorem reformulates Iyama's question in terms of the vanishing of Ext; and highlights its relation with the rigidity of filtered colimits of $\mathcal{M}$. Also, we show that Add$(\mathcal{M})$ is an n-cluster tilting subcategory of $\Lambda$-Mod if and only if Add$(\mathcal{M})$ is a maximal n-rigid subcategory of $\Lambda$-Mod if and only if $\lbrace X\in \Lambda$-Mod$~|~ {\rm Ext}^i_{\Lambda}(\mathcal{M},X)=0 ~~~ {\rm for ~all}~ 0

Let $\Lambda$ be a finite dimensional algebra. In this paper we show that there is a natural bijection between cosilting modules in Mod$\Lambda$ and semibricks in Mod$\Lambda$ satisfying some condition. Also this bijection restricts to a bijection between all semibricks in mod$\Lambda$ and a certain subclass of cosilting modules. These bijections are generalizations of Asai's correspondence [7] between support $\tau^-$-tilting modules and right finite semibricks.

Let $A$ be a finite-dimensional algebra, and $\mathfrak{M}$ be a $d$-cluster tilting subcategory of mod$A$. From the viewpoint of higher homological algebra, a natural question to ask is when $\mathfrak{M}$ induces a $d$-cluster tilting subcategory in Mod$A$. In this paper, we investigate this question in a more general form. Let $\mathcal{M}$ be a small $d$-abelian category of an abelian category $\mathcal{A}$. The completion of $\mathcal{M}$, denoted by Ind$(\mathcal{M})$, is defined as the universal completion of $\mathcal{M}$ with respect to filtered colimits. We explore Ind$(\mathcal{M})$ and demonstrate its equivalence to the full subcategory $\mathcal{L}_d(\mathcal{M})$ of Mod$\mathcal{M}$, comprising left $d$-exact functors. Notably, while Ind$(\mathcal{M})$ as a subcategory of $\frac{Mod\mathcal{M}}{Eff(\mathcal{M})}$, 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 $\mathfrak{M}$ of mod$A$, $\overrightarrow{\mathfrak{M}}$, consists of all filtered colimits of objects from $\mathfrak{M}$, is a generating-cogenerating, functorially finite subcategory of Mod$A$. The question of whether $\mathfrak{M}$ 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 $\overrightarrow{\mathfrak{M}}$ is a $2$-cluster tilting subcategory if and only if $\mathfrak{M}$ is of finite type. Thus, the question regarding whether $\overrightarrow{\mathfrak{M}}$ is a $d$-cluster tilting subcategory of Mod$ A$ appears to be equivalent to the Iyama's qestion about the finiteness of $\mathfrak{M}$.

Let $\Lambda$ be an artin algebra and $\mathcal{C}$ be a functorially finite subcategory of mod$\Lambda$ which contains $\Lambda$ or $D\Lambda$. We use the concept of the infinite radical of $\mathcal{C}$ and show that $\mathcal{C}$ has an additive generator if and only if rad$^\infty_{\mathcal{C}}$ vanishes. In this case we describe the morphisms in powers of the radical of $\mathcal{C}$ in terms of its irreducible morphisms. Moreover, under a mild assumption, we prove that $\mathcal{C}$ is of finite representation type if and only if any family of monomorphisms (epimorphisms) between indecomposable objects in $\mathcal{C}$ is noetherian (conoetherian). Also, by using injective envelopes, projective covers, left $\mathcal{C}$-approximations and right $\mathcal{C}$-approximations of simple $\Lambda$-modules, we give other criteria to describe whether $\mathcal{C}$ is of finite representation type. In addition, we give a nilpotency index of the radical of $\mathcal{C}$ which is independent from the maximal length of indecomposable $\Lambda$-modules in $\mathcal{C}$., Comment: We changed the title and some results of the paper. We also added some new results

In this paper, by using functor rings and functor categories, we study finiteness and purity of subcategories of the module categories. We give a characterisation of contravariantly finite resolving subcategories of the module category of finite representation type in terms of their functor rings. We also characterize contravariantly finite resolving subcategories of the module category $\Lambda${\rm -mod} that contain the Jacobson radical of $\Lambda$ of finite type, by their functor categories. We study the pure semisimplicity conjecture for a locally finitely presented category ${\underrightarrow{\lim}}\mathscr{X}$ when $\mathscr{X}$ is a covariantly finite subcategory of $\Lambda$-mod and every simple object in Mod$(\mathscr{X}^{\rm op})$ is finitely presented and give a characterization of covariantly finite subcategories of finite representation type in terms of decomposition properties of their closure under filtered colimits. As a consequence we study finiteness and purity of $n$-cluster tilting subcategories and the subcategory of the Gorenstein projective modules of the module categories. These results extend and unify some known results.

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.

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., Comment: arXiv admin note: text overlap with arXiv:2011.15107 by other authors

Let $\Lambda$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of mod$\Lambda$. We show that $\mathcal{M}$ has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of $\mathcal{M}$ if and only if every effaceable functor $\mathcal{M}\rightarrow Ab$ has finite length. As a consequence we show that if mod$\Lambda$ 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 $\Lambda$.

We study the Auslander ring of a basic left K\"othe ring $\Lambda$ and give a characterization of basic left K\"othe rings in terms of their Auslander rings. We also study the functor category Mod$((\Lambda$-mod$)^{\rm op})$ and characterize basic left K\"othe rings $\Lambda$ by using functor categories Mod$((\Lambda$-mod$)^{\rm op})$. As a consequence we show that there exists a bijection between the Morita equivalence classes of left Kawada rings and the Morita equivalence classes of Auslander generalized right QF-2 rings.

We study Morita equivalence and Morita duality for rings with local units. We extend the 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.

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 $\widetilde{A}_n$ is equal to the corresponding lower bound cluster algebra generated by projective cluster variables.

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of finitely presented functors $mod$-$\mathcal{M}$ modulo the subcategory of effaceable functors $mod_0$-$\mathcal{M}$ has an $n$-cluster tilting subcategory which is equivalent to $\mathcal{M}$. This gives a higher-dimensional version of Auslander's formula., Comment: arXiv admin note: text overlap with arXiv:2001.01254

Let $\mathcal{M}$ be a small $n$-abelian category. We show that the category of absolutely pure group valued functors over $\mathcal{M}$, denote by $\mathcal{L}_2(\mathcal{M},\mathcal{G})$, is an abelian category and $\mathcal{M}$ is equivalent to a full subcategory of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ in such a way that $n$-kernels and $n$-cokernels are precisely exact sequences of $\mathcal{L}_2(\mathcal{M},\mathcal{G})$ with terms in $\mathcal{M}$. This gives a higher-dimensional version of the Freyd-Mitchell embedding theorem for $n$-abelian categories.

From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $\Lambda$ be an Artin algebra and $\mathcal{M}$ be an $n$-cluster tilting subcategory of $Mod$-$\Lambda$. We show that $\mathcal{M}$ is pure semisimple if and only if each module in $\mathcal{M}$ is a direct sum of finitely generated modules. Let $\mathfrak{m}$ be an $n$-cluster tilting subcategory of $mod$-$\Lambda$. We show that $Add(\mathfrak{m})$ is an $n$-cluster tilting subcategory of $Mod$-$\Lambda$ if and only if $\mathfrak{m}$ has an additive generator if and only if $Mod(\mathfrak{m})$ is locally finite. This generalizes Auslander's classical results on pure semisimplicity of Artin algebras., Comment: to appear in Journal of Algebra

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

We characterize right $4$-Nakayama artin algebras which appear naturally in the study of representation-finite artin algebras. For a right $4$-Nakayama artin algebra $\Lambda$, we classify all finitely generated indecomposable right $\Lambda$-modules and then we compute all almost split sequences over $\Lambda$. We also give a characterization of right $4$-Nakayama finite dimensional $K$-algebras in terms of their quivers with relations., Comment: arXiv admin note: substantial text overlap with arXiv:1805.04412

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.

Let $\Lambda$ 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 $\Lambda$ form a maximal forward hom-orthogonal sequence of $\Lambda$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $\Lambda$. Thus we give a proof of Igusa-Todorov's conjecture.

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

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 $\Lambda$ is representation-finite if and only if $\Lambda$ 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 $\Lambda$ is right $n$-coNakayama if and only if $\Lambda$ 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., Comment: to appear in Algebr. Represent. Theory

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., Comment: to appear in J. Australian Math. Soc

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 cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.

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., Comment: Forum Math. (27)2015. arXiv admin note: text overlap with arXiv:1207.3466 by other authors

We prove that the Crisp and Gow's quiver operation on a finite quiver Q produces a new quiver Q' with fewer vertices, such that the finite dimensional algebras kQ/J^2 and kQ'/J^2 are singularly equivalent. This operation is a general quiver operation which includes as specific examples some operations which arise naturally in symbolic dynamics (e.g., (elementary) strong shift equivalent, (in-out) splitting, source elimination, etc.)., Comment: to appear in J. Pure Appl. Algebra

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

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., Comment: 19 pages; revised version; to appear in Journal of Algebra