Tran Do Minh Chau’s research while affiliated with Thai Nguyen University and other places

Let $(R,\mathfrak {m})$ be a Noetherian local ring, let I be an ideal of R, and let M be a finitely generated R-module with $d=\dim (M)$. In this paper, we establish shifted principles under localization and completion for attached primes of the top local cohomology module ${H^{d}_{I}}(M).$ We characterize the catenarity, the weak going-up property, and the strong Lichstenbaum-Hartshorne vanishing property of the base ring R in terms of these shifted principles of the top local cohomology modules.

Let [Formula: see text] be a homogeneous graded ring, where [Formula: see text] is a Noetherian local ring. Let [Formula: see text] be a finitely generated graded [Formula: see text]-module. For [Formula: see text] set [Formula: see text]. Denote by [Formula: see text] the set of all prime ideals of [Formula: see text] containing [Formula: see text]. For [Formula: see text], let [Formula: see text] be the set of all [Formula: see text] such that [Formula: see text] In this paper, we prove that the sets [Formula: see text] and [Formula: see text] do not depend on [Formula: see text] for [Formula: see text]. We show that the annihilators [Formula: see text], [Formula: see text] are eventually stable, where [Formula: see text] for [Formula: see text]. As an application, we prove the asymptotic stability of some loci contained in the non-Cohen–Macaulay locus of [Formula: see text].

The notion of 2-AGL ring in dimension one which is a natural generalization of almost Gorenstein local ring is posed in terms of the rank of Sally modules of canonical ideals. The basic theory is developed, investigating also the case where the rings considered are numerical semigroup rings over fields. Examples are explored.

Let M be a finitely generated module over a Noetherian local ring R. In this paper we introduce the notion of sequential polynomial type of M, which is denoted by , in order to measure how far M is different from the sequential Cohen–Macaulayness. We study the sequential polynomial type under localization and -adic completion. We investigate an ascent–descent property of sequential polynomial type between M and for certain parameter x of M. When R is a quotient of a Gorenstein local ring, we describe in term of the deficiency modules of M.

Let (R, 𝔪) be a Noetherian local ring. Denote by N-dimRA the Noetherian dimension of an Artinian R-module A. In this paper, we give some characterizations for the ring R to satisfy N-dimRA =dim (R/AnnRA) for certain Artinian R-modules A. Then the existence of a co-localization compatible with Artinian R-modules is studied and it is shown that if it is compatible with local cohomologies of finitely generated modules, then the base ring is universally catenary and all of its formal fibers are Cohen-Macaulay.

Let (R,m) be a Noetherian local ring and M a finitely generated R-module. It is well known that the local cohomology module Hmi(M) is Artinian for all i ≥ 0. Following I.G. Macdonald [8], denote by AttRHmi(M) the set of attached primes of Hmi(M). This paper is concerned with clarifying the structure of the base ring R via a relation between AttRHmi(M) and AttR̂Hmi(M). Some characterizations for R being universally catenary with all Cohen-Macaulay formal fibers are given.

Let (R,m) be a Noetherian local ring and I an ideal of R. Let M be a finitely generated R-module with dim. M=d. It is clear by Matlis duality that if R is complete then HId(M) satisfies the following property: AnnR(0:HId(M)p) = p for all prime ideals p ⊇ AnnRHId(M). However, HId(M) does not satisfy the property (*) in general. In this paper we characterize the property (*) of HId(M) in order to study the catenarity of the ring R/AnnRHId(M), the set of attached primes AttRHId(M), the co-support CosR(HId(M)), and the multiplicity of HId(M). We also show that if HId(M) satisfies the property (*) then HId(M)≅Hmd(M/N) for some submodule N of M.