In the theory of modules over commutative rings there are several possibilities of defining associated prime ideals. The usual definition of an associated prime ideal p for a module M is that p is the annihilator of an element of M . In [2] §1 exercise 17 a generalization of this notion is given. p is called weakly associated (faiblement associé) to M if p is minimal in the set of the prime ide...

s. 1 Let (R,m) be a Noetherian local ring which is a quotient of a Gorenstein local ring. Let M be a finitely generated R-module. In this paper, we study the structure of the canonical module K(RnM) of the idealization RnM via the polynomial type introduced by N. T. Cuong [5]. In particular, we give a characterization for K(RnM) being Cohen-Macaulay and generalized Cohen-Macaulay.

Let $R$ be a commutative Noetherian ring with non-zero identity and $fa$ an ideal of $R$. Let $M$ be a finite $R$--module of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properti...

It is proved that if R is a right and left Noetherian ring then the right R-module RI satisfies the ascending chain condition on n-generated submodules, for every positive integer n. Mathematics Subject Classification (2000): 16P70

In this note, it is proved that over a commutative noetherian henselian non-Gorenstein local ring there are infinitely many isomorphism classes of indecomposable totally reflexive modules, if there is a nonfree cyclic totally reflexive module.

Let R be a Noetherian local ring and Ω an arbitrary R-module of finite depth and finite projective dimension. The flat dimension of Ω is at least depth(R)−depth(Ω) with equality in the following cases: (i) Ω is finitely generated over some Noetherian local R-algebra S; (ii) dim(R) = 1; (iii) dim(R) = 2 and Ω is separated; (iv) R is CohenMacaulay, dim(R) = 3 and Ω is complete.

We introduce and investigate the notion of GC -projective modules over (possibly non-noetherian) commutative rings, where C is a semidualizing module. This extends Holm and Jørgensen’s notion of C-Gorenstein projective modules to the non-noetherian setting and generalizes projective and Gorenstein projective modules within this setting. We then study the resulting modules of finite GC-projectiv...

Let R <-* S be an embedding of associative noetherian rings such that 5 is finitely generated as a right Ä-module. There is a correspondence from the prime spectrum of S to the prime spectrum of R obtained by associating to a given prime ideal P of S the prime ideals of R minimal over P n R . The prime and primitive ideal theories for several specific noncommutative noetherian rings, including ...

Let R be a commutative ring. An R-module M is called co-multiplication provided that foreach submodule N of M there exists an ideal I of R such that N = (0 : I). In this paper weshow that co-multiplication modules are a generalization of strongly duo modules. Uniserialmodules of finite length and hence valuation Artinian rings are some distinguished classes ofco-multiplication rings. In additio...

In the study of hereditary Noetherian rings, it is clear that hereditary Noetherian prime rings will play a central role (see, for example, [12]). Here we study the (two-sided) ideals of an hereditary Xoetherian prime ring and, as a consequence, ascertain the structure of factor rings and torsion modules. The torsion theory represents a generalization of similar results about Dedekind prime rin...