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Let S be an algebra over a commutative ring R, and let σ be an automorphism on S. In this paper, we investigate the notion of generalized σ-derivations on modules, which is an extension of generalized derivations on modules introduced by Nakajima. Namely, we study homological properties of generalized σ-derivations. Also, we equip the category of functors that send S/R-modules to R-modules of generalized σ-derivations with a tensor product. We show this category is semi-monoidal. As an application, we characterize when a generalized derivation on a path algebra of an acyclic quiver is a generalized inner derivation. [ABSTRACT FROM AUTHOR]

For a finite free and projective EI category, we prove that Gorenstein-projective modules over its category algebra are closed under the tensor product if and only if each morphism in the given category is a monomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we present the categories associated to the square matrix with coefficients 2 of order 3. The family of categories associated to these matrices has five isomorphism classes. In the case of a matrix of order higher than 3, we only demonstrate upper and lower bounds for the number of associated categories. [ABSTRACT FROM AUTHOR]

We introduce and investigate weak relative Rickart objects and dual weak relative Rickart objects in abelian categories. Several types of abelian categories are characterized in terms of (dual) weak relative Rickart properties. We relate our theory to the study of relative regular objects and (dual) relative Baer objects. We also give some applications to module and comodule categories. [ABSTRACT FROM AUTHOR]

We study how to read from the ordinary quiver of a representation-finite string algebra, the minimal lower boundm≥ 1 such that themth power of the radical of its module category vanishes. We also show how to read the degree of any irreducible morphism in such a representation-finite string algebras considering strings. [ABSTRACT FROM PUBLISHER]

Let (R,𝔪) be a local ring andIan ideal. The aim of the present paper is twofold. At first we continue the investigation to comparefgrade(I,R) withdepth R∕Iand further we derive some results on the vanishing of Lyubeznik numbers. [ABSTRACT FROM PUBLISHER]

A set valued representation of the Kronecker quiver is nothing but a quiver. We apply the forgetful functor from vector spaces to sets and compare linear with set valued representations of the Kronecker quiver. [ABSTRACT FROM PUBLISHER]

We study the projective spectrum of the Rees algebra of a module and characterize it by a universal property. As applications, we give descriptions of universal flatifications of modules and of birational projective morphisms. [ABSTRACT FROM AUTHOR]

For anR-moduleM, projective in σ[M] and satisfying ascending chain condition on left annihilators, the authors introduce the concept of Goldie module. The authors also use the concept of semiprime module defined by Raggi et al. in [15] to give necessary and sufficient conditions for anR-moduleM, to be a semiprime Goldie module. This theorem is a generalization of Goldie's theorem for semiprime left Goldie rings. Moreover, the authors prove thatMis a semiprime (prime) Goldie module if and only if the ringS = EndR(M) is a semiprime (prime) right Goldie ring. Also, we study the case whenMis a duo module. [ABSTRACT FROM PUBLISHER]

We give an alternative proof to the fact that, if the square of the infinite radical of the module category of an Artin algebra is equal to zero, then the algebra is of finite type by making use of the theory of postprojective and preinjective partitions. Further, we use this new approach in order to get a characterization of finite subcategories of Δ-good modules of a quasi-hereditary algebra in terms of depth of morphisms similar to a recently obtained characterization of Artin algebras of finite type. [ABSTRACT FROM PUBLISHER]

In [9] Giraldo and Merklen studied irreducible morphisms in the categories 𝒞(𝒜) and D−(Λ), where 𝒞(𝒜) is the category of complexes over an abelian Krull–Schmidt category 𝒜 and D−(Λ) is the derived category of the bounded above complexes of finite generate left modules, over an Artin algebra Λ. In this work, we continue the study of irreducible morphism having one finite irreducible truncation. [ABSTRACT FROM AUTHOR]

We generalize results on existence of recollement situations of singularity categories of lower triangular Gorenstein algebras and stable monomorphism categories of Cohen–Macaulay modules. [ABSTRACT FROM AUTHOR]