In this paper, we investigate the notions of X ⊥ -projective, id="M2"> -injective, and id="M3"> -flat modules give some characterizations these modules, where id="M4"> is a class left modules. We prove that all id="M5"> -projective Kaplansky. Further, if id="M6"...

Motivated by the relation between Schur algebra and group of a symmetric group, along with other similar examples in algebraic Lie theory, Min Fang Steffen Koenig [15], [16] addressed some behaviour endomorphism generator over algebra, which they called gendo-symmetric algebra. Continuing this line works, we classify article representation-finite algebras that have precisely one isomorphism cla...

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the ap...

Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective ...

An R-module is called semi-endosimple if it has no proper fully invariant essential submodules. For a quasi-projective retractable module MR we show that M is finitely generated semi-endosimple if and only if the endomorphism ring of M is a finite direct sum of simple rings. For an arbitrary module M , conditions equivalent to the semi-endosimplicity of its quasi-injective hull are found. As co...

For an (n− 1)-Auslander algebra Λ with global dimension n, we give some necessary conditions for Λ admitting a maximal (n − 1)-orthogonal subcategory in terms of the properties of simple Λ-modules with projective dimension n − 1 or n. For an almost hereditary algebra Λ with global dimension 2, we prove that Λ admits a maximal 1orthogonal subcategory if and only if for any non-projective indecom...

The relative homotopy theory of modules, including the (module) homotopy exact sequence, was developed by Peter Hilton (1965). Our thrust is to produce an alternative proof of the existence of the injective homotopy exact sequence with no reference to elements of sets, so that one can define the necessary homotopy concepts in arbitrary abelian categories with enough injectives and projectives, ...

In this paper we give characterizations of essential left ideals of a C*-algebra A in terms of their properties as operator A-modules. Conversely, we seek C*-algebraic characterizations of those ideals J in A such that A is an essential extension of J in various categories of operator modules. In the case of two-sided ideals, we prove that all the above concepts coincide. We obtain results, ana...