In ring and module theory, one concept is the projective module. A said to be if it a direct sum of independent modules. (U, R) an approximation space with non-empty set equivalence relation If X subset U, we can form upper lower approximation. rough Apr(X) not equal under Apr(X). The theory applies algebraic structures, including groups, rings, modules, homomorphisms. this study, will investig...

It was proved by Beligiannis and Krause that over certain Artin algebras, there are Gorenstein flat modules which are not direct limits of finitely generated Gorenstein projective modules. That is, these algebras have no Gorenstein analogue of the Govorov-Lazard Theorem. We show that, in fact, there is a large class of rings without such an analogue. Namely, let R be a commutative local noether...

Let G be a finite group, and let Ω := {t ∈ G | t = 1}. Then Ω is a G-set under conjugation. Let k be an algebraically closed field of characteristic 2. It is shown that each projective indecomposable summand of the G-permutation module kΩ is irreducible and self-dual, whence it belongs to a real 2-block of defect zero. This, together with the fact that each irreducible kG-module that belongs to...

To a B-coring and a (B, A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projec...

Introduction. A module over a ring will be said to be locally projective if and only if every finitely generated submodule is projective. As will be shown (7.14), it readily follows from known facts that if M is a locally projective module over a regular ring R, then the set L(M, R) of all finitely generated submodules of M is a relatively complemented modular lattice. This paper is concerned w...

Let Λ be an Auslander’s 1-Gorenstein Artinian algebra with global dimension 2. If Λ admits a trivial maximal 1-orthogonal subcategory of modΛ, then for any indecomposable module M ∈ modΛ, we have that the projective dimension of M is equal to 1 if and only if so is its injective dimension and that M is injective if the projective dimension of M is equal to 2, which implies that Λ is almost here...

We will introduce the notion of Gorenstein category as the convenient setup for doing Gorenstein Homological Algebra in categories of sheaves, or in general in categories without enough projective objects. We will illustrate this notion by showing that the category of Qcoh(X) of quasi-coherent sheaves on a locally Gorenstein projective scheme fits into this setup. Then we will focus on the cate...

In this paper, we introduce a novel pictorial approach for solving problems in n-dimensional Euclidean spaces called the n-dimensional projective approach. The projective approach is based on a hierarchical and modular architecture, where its ground module is rooted on geometrical concepts. The result is an effective and consistent spatial approach able to solve problems in ndimensional spaces ...