Let $\mathfrak {a} $ be an ideal of a commutative Noetherian ring $ R $ and $ M $ a finitely
generated $ R $-module. Let $ t $ be a natural integer. It is shown that there is a finite subset …

Let 𝑅 be a commutative Noetherian local ring with residue field 𝑘. We show that, if a finite
direct sum of syzygy modules of 𝑘 maps onto'a semidualizing module'or'a non-zero maximal …

In this paper we prove the following generalization of a result of Hartshorne: Let $ T $ be a
commutative Noetherian local ring of dimension at least two, $ R= T [x_1,\dots, x_n] $, and …

Let R be ad-dimensional regular local ring, I an ideal of R, and M a finitely generated R-
module of dimension n. We prove that the set of associated primes of Ext Ri (R/I, H Ij (M)) is …

Let R be a commutative noetherian local ring with residue field k. We introduce two new
invariants of R-modules and compute them for the module k. As a consequence, we show …

Let (R, 𝔪) be a complete local ring, 𝔞 an ideal of R, and M a finitely generated R-module.
The aim of this paper is to show that for any non-negative integer n,, where is the n th …

A module is said to be extending, if every closed (ie complement) submodule is a direct
summand. This property is usually denoted by (Q). It is, obviously, equivalent to the …

Let $(R,{\mathfrak {m}}) $ be a commutative Noetherian local ring, and let $ I $ and $ J $ be
two proper ideals of $ R $. Let $ M $ be a non-zero finitely generated $ R-$ module. We …

Let 𝔞 be an ideal of a commutative Noetherian ring R with identity and let M and N be two
finitely generated R-modules. Let t be a positive integer. It is shown that is contained in the …

Let $ R $ be a commutative Noetherian local ring. We show that $ R $ is Gorenstein if and
only if every finitely generated $ R $-module can be embedded in a finitely generated $ R …