Let k be a commutative ring, A a k-algebra, C an A-coring that is projective as a left A-module, ∗C the dual ring of C and Λ a right C-comodule that is finitely generated as a left ∗C-module. We give necessary and sufficient conditions for projectivity and flatness of a module over the endomorphism ring EndC(Λ). If C contains a grouplike element, we can replace Λ with A.

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a cyclic p-group is permutation projective.

Let (A, m, k) be a complete intersection of codimension c, and k̃ the algebraic closure of k. We show that every homogeneous algebraic subset of k̃c is the cohomological support variety of an A-module, and that the projective variety of a complete indecomposable maximal Cohen-Macaulay A-module is connected.

In this paper we use "ring changed'' Gorenstein homologicaldimensions to define Cohen-Macaulay injective, projective and flatdimensions. For doing this we use the amalgamated duplication of thebase ring with semi-dualizing ideals. Among other results, we prove that finiteness of these new dimensions characterizes Cohen-Macaulay rings with dualizing ideals.

let r be a right gf-closed ring with finite left and right gorenstein global dimension. we prove that if i is an ideal of r such that r/i is a semi-simple ring, then the gorensntein flat dimensnion of r/i as a right r-module and the gorensntein injective dimensnnion of r/i as a left r-module are identical. in particular, we show that for a simple module s over a commutative gorensntein ring r, ...

In this paper, we introduce the “semi-topological K-homology” of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasiprojective complex variety Y coincides with the connective topological Khomology of the associated anal...

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...