For any ideal I in a Noetherian local ring or graded standard K-algebra over field K, we introduce the socle module Soc(I), whose components give us of powers I. It is observed that Soc(I) finitely generated fiber cone In case S polynomial and all I⊆S have linear resolution, define Soc∗(I), which Rees edge graph for classes polymatroidal ideals, study structure their modules.

Abstract A module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of such that / finitely generated, there exists a direct summand V $$V \subseteq U$$ V ⊆ U and generated. It shown non-finitely generated fr...

Let M be a module over Noetherian local ring A. We study M-independent sequences of elements the maximal ideal mA in sense Lech and Hanes. The main tool is new characterization M-independence sequence terms associated Koszul complex. As applications, we give result linkage theory, freeness criterion for existence strongly length equal to embedding dimension edim(A) A, another inspired from patc...

The theory of local homology modules was initiated by Matlis in 1974. It is a dual version of the theory of local cohomology modules. Mohammadi and Divaani-Aazar (2012) studied the connection between local homology and Gorenstein flat modules by using Gorenstein flat resolutions. In this paper, we introduce generalized local homology modules for complexes and we give several ways for computing ...

Let R be a noetherian ring and M a finite R-module. With a linear form χ on M one associates the Koszul complex K(χ). If M is a free module, then the homology of K(χ) is well-understood, and in particular it is grade sensitive with respect to Imχ. In this note we investigate the case of a module M of projective dimension 1 (more precisely, M has a free resolution of length 1) for which the firs...

Let R be a noetherian ring andM a finite R-module. With a linear form χ onM one associates the Koszul complex K(χ). IfM is a free module, then the homology of K(χ) is well-understood, and in particular it is grade sensitive with respect to Imχ. In this note we investigate the case of a module M of projective dimension 1 (more precisely, M has a free resolution of length 1) for which the first n...

let r be a ring,  be an endomorphism of r and mr be a -rigid module. amodule mr is called quasi-baer if the right annihilator of a principal submodule of r isgenerated by an idempotent. it is shown that an r-module mr is a quasi-baer module if andonly if m[[x]] is a quasi-baer module over the skew power series ring r[[x; ]].

We first provide an example of a ring R such that all possible 2 × 2 structural matrix rings over R are isomorphic. However, we prove that the underlying graphs of any two isomorphic structural matrix rings over a semiprime Noetherian ring are isomorphic, i.e. the underlying Boolean matrix B of a structural matrix ring M(B,R) over a semiprime Noetherian ring R can be recovered, contrary to the ...

Abstract. Let (R,P) be a Noetherian unique factorization do-main (UFD) and M be a finitely generated R-module. Let I(M)be the first nonzero Fitting ideal of M and the order of M, denotedord_R(M), be the largest integer n such that I(M) ⊆ P^n. In thispaper, we show that if M is a module of order one, then either Mis isomorphic with direct sum of a free module and a cyclic moduleor M is isomorphi...

Let  be a local Cohen-Macaulay ring with infinite residue field,  an Cohen - Macaulay module and  an ideal of  Consider  and , respectively, the Rees Algebra and associated graded ring of , and denote by  the analytic spread of  Burch’s inequality says that  and equality holds if  is Cohen-Macaulay. Thus, in that case one can compute the depth of associated graded ring of  as  In this paper we ...