arXiv subjects
Publications and source records attributed to Tran Nguyen An.
Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of $\mathfrak{m} \times M$-primary ideal of the idealization. This generalizes one of the main results of S.Goto et al. (see \cite[Theorem 2.2]{GKL}).
Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ a finitely generated $R$-module with $\dim_R(M)=d$. Denote by $\depth_R(I,M)$ the depth of $M$ in $I$. In \cite{HT}, C. Huneke and V. Trivedi proved that if $R$ is a quotient of a regular ring then there exists a finite subset $Λ_M $ of $\Spec(R)$ such that $$\depth_R(I,M)=\underset{\p\in Λ_M}{\min} \big\{ \depth_{R_{\p}}(M_{\p})+ \docao\big((I+\p)/\p\big) \big\}.$$ Denote by $\Psupp^i_R(M)=\{\frak p\in\Spec(R)\mid H^{i-\dim(R/\frak p)}_{\frak p R_{\frak p}}(M_{\frak p})\neq 0\}$ the $i$-th pseudo support of $M$ defined by M. Brodmann and R. Y. Sharp \cite{BS1}. In this paper, we prove that if $\Psupp^i_R(M)$ is closed for all $i\leq d$ then the above formula of $\depth_R(I,M)$ holds true, where $Λ_M =\underset{0\leq i\leq d}{\bigcup} \min \Psupp^i_R(M)$. In particular, if $R$ is a quotient of a Cohen-Macaulay local ring then $Λ_M =\underset{0\leq i\leq d}{\bigcup}\min\Var\big(\Ann_R(H_{\m}^i(M))\big)$. We also give some examples to clarify the results.
A foundational result by C. Huneke and V. Trivedi provides a formula for the depth of an ideal in terms of height, computed over a finite set of prime ideals, for rings that are homomorphic images of regular rings. Building on a result by the first author for local quotients of Cohen-Macaulay rings, this paper first gives a new proof and derives a similar formula for the finiteness dimension. Our main result then establishes the depth formula for non-local rings that are homomorphic images of a finite-dimensional Gorenstein ring.
In this paper, we give a relation between the Hilbert multiplicity and the irreducible multiplicity. As an application, we characterize Ulrich modules in term of the irreducible multiplicity.
Let $R$ be a Noetherian ring. For a finitely generated $R$-module $M$, Northcott introduced the reducibility index of $M$, which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule $0$ in $M$. On the other hand, for an Artinian $R$-module $A$, Macdonald proved that the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation of $A$ does not depend on the choice of the representation. This number is called the sum-reducibility index of $A$. In the former part of this paper, we compute the reducibility index of $S\otimes_R M$, where $R\to S$ is a flat homomorphism of Noetherian rings. Especially, the localization, the polynomial extension, and the completion of $R$ are studied. For the latter part of this paper, we clarify the relation among the reducibility index of $M$, that of the completion of $M$, and the sum-reducibility index of the Matlis dual of $M$.
This paper studies the question of when the Rees algebras associated to arbitrary filtration of ideals are sequentially Cohen-Macaulay. Although this problem has been already investigated by N. T. Cuong, S. Goto and H. L. Truong, their situation is quite a bit of restricted, so we are eager to try the generalization of their results.
In this paper, we study the two different topics related to sequentially Cohen-Macaulay modules. The questions are when the sequentially Cohen-Macaulay property preserve the localization and the module-finite extension of rings.