• Journal of the Chungcheong Mathematical Society
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• v.27 no.4
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• pp.625-633
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• 2014

We study some results which concern the types of Noetherian local rings, and improve slightly the previous result: For a complete unmixed (or quasi-unmixed) Noetherian local ring A, we prove that if either $A_p$ is Cohen-Macaulay, or $r(Ap){\leq}depth$ $A_p+1$ for every prime ideal p in A, then A is Cohen-Macaulay. Also, some analogous results for modules are considered.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.519-530
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• 2014

Let (R,m) be a Noetherian local ring and M a finitely generated R-module. For an integer s > -1, we say that M is Cohen-Macaulay in dimension > s if every system of parameters of M is an M-sequence in dimension > s introduced by Brodmann-Nhan [1]. In this paper, we give some characterizations for Cohen-Macaulay modules in dimension > s in terms of the Noetherian dimension of the local cohomology modules $H^i_m(M)$, the polynomial type of M introduced by Cuong [5] and the multiplicity e($\underline{x}$;M) of M with respect to a system of parameters $\underline{x}$.

• Journal of Integrative Natural Science
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• v.11 no.2
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• pp.82-86
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• 2018

We show that the (Castelnuovo-Mumford) regularity of a homogeneous Cohen-Macaulay ring agrees with some of the invariants defined the papers, and we study a ring of small index.

• Bulletin of the Korean Mathematical Society
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• v.39 no.4
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• pp.577-587
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• 2002

Let A be a commutative ring and M an Artinian .A-module. Let $\sigma$ be a torsion radical functor and (T, F) it's corresponding partition of Spec(A) In [1] the concept of Cohen-Macauly modules was generalized . In this paper we shall define $\sigma$-co-Cohen-Macaulay (abbr. $\sigma$-co-CM). Indeed this is one of the aims of this paper, we obtain some satisfactory properties of such modules. An-other aim of this paper is to generalize the concept of cograde by using the left derived functor $U^{\alpha}$$_{I}$(-) of the $\alpha$-adic completion functor, where a is contained in Jacobson radical of A.A.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1323-1330
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• 2017

For a nondegenerate projective irreducible variety $X{\subset}{\mathbb{P}}^r$, it is a natural problem to find an upper bound for the value of $${\ell}_{\beta}(X)=max\{length(X{\cap}L){\mid}L={\mathbb{P}}^{\beta}{\subset}{\mathbb{P}}^r,\;{\dim}(X{\cap}L)=0\}$$ for each $1{\leq}{\beta}{\leq}e$. When X is locally Cohen-Macaulay, A. Noma in [10] proves that ${\ell}_{\beta}(X)$ is at most $d-e+{\beta}$ where d and e are respectively the degree and the codimension of X. In this paper, we construct some surfaces $S{\subset}{\mathbb{P}}^5$ of degree $d{\in}\{7,{\ldots},12\}$ which satisfies the inequality $${\ell}_2(S){\geq}d-3+{\lfloor}{\frac{d}{2}}{\rfloor}$$. This shows that Noma's bound is no more valid for locally non-Cohen-Macaulay varieties.

• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.210-214
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• 1994

The purpose of this paper is to investigate the relationship between the depths of the Rees algebra R[It] and the associated graded ring g $r_{I}$(R) of an ideal I in a local ring (R,m) of dim(R) > 0. The relationship between the Cohen-Macaulayness of these two rings has been studied extensively. Let (R, m) be a local ring and I an ideal of R. An ideal J contained in I is called a reduction of I if J $I^{n}$ = $I^{n+1}$ for some integer n.geq.0. A reduction J of I is called a minimal reduction of I. The reduction number of I with respect to J is defined by (Fig.) S. Goto and Y.Shimoda characterized the Cohen-Macaulay property of the Rees algebra of the maximal ideal of a Cohen-Macaulay local ring in terms of the Cohen-Macaulay property of the associated graded ring of the maximal ideal and the reduction number of that maximal ideal. Let us state their theorem.m.m.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.545-551
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• 2009

We investigate the behavior of maps in minimal injective resolution of an A-module M when ${\mu}_t$(m,M) = 1 for some t, and we develop slightly the fact that a module of type 1 is Cohen-Macaulay.

• Journal of the Korean Mathematical Society
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• v.43 no.4
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• pp.871-883
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• 2006

We show that the Auslander index is the same as the column invariant over Gorenstein local rings. We also show that Ding's conjecture ([13]) holds for an isolated non-Gorenstein ring A satisfying a certain condition which seems to be weaker than the condition that the associated graded ring of A is Cohen-Macaulay.

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.275-280
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• 2020

Let (R, m) be a Noetherian local Cohen-Macaulay ring and I be a proper ideal of R. Assume that β_R(I, R) denotes the constant value of depth_R(R/Iⁿ) for n ≫ 0. In this paper we introduce the new notion γ_R(I, R) and then we prove the following inequalities: β_R(I, R) ≤ γ_R(I, R) ≤ dim R - cd(I, R) ≤ dim R/I. Also, some applications of these inequalities will be included.

• Honam Mathematical Journal
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• v.20 no.1
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• pp.1-14
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• 1998

In this short note we study the torsion theories over a commutative ring R and discuss a relative dimension related to such theories for R-modules. Let ${\sigma}$ be a torsion functor and (T, F) be its corresponding partition of Spec(R). The concept of ${\sigma}$-Cohen Macaulay (abbr. ${\sigma}$-CM) module is defined and some of the main points concerning the usual Cohen-Macaulay modules are extended. In particular it is shown that if M is a non-zero ${\sigma}$-CM module over R and S is a multiplicatively closed subset of R such that, for all minimal element of T, $S{\cap}p={\emptyset}$, then $S^{-1}M$ is a $S^{-1}{\sigma}$-CM module over $S^{-1}$R, where $S^{-1}{\sigma}$ is the direct image of ${\sigma}$ under the natural ring homomorphism $R{\longrightarrow}S^{-1}R$.