• Bulletin of the Korean Mathematical Society
• /
• v.44 no.4
• /
• pp.871-878
• /
• 2007

We define a numerical invariant $row_{CM}(A)$ over Cohen-Macaulay local ring A, which is related to rows of the presenting matrices of maximal Cohen-Macaulay modules without free summands. We show that $row(A)=row_{CM}(A)$ for a Cohen-Macaulay(not necessarily Gorenstein) local ring A.

• Journal of Integrative Natural Science
• /
• v.7 no.4
• /
• pp.278-282
• /
• 2014

We define a numerical invariant $row^*_j(A)$ over Cohen-Macaulay local ring A, which is related to the presenting matrices of the j-th syzygy module (with or without free summands). We show that $row_d(A)$=$row_{CM}(A)$ and $row^*_d(A)$=$row^*_{CM}(A)$ for a Cohen-Macaulay local ring A of dimension d.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.5
• /
• pp.989-996
• /
• 2012

Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper we study the amalgamated duplication ring $R{\bowtie}I$ which is introduced by D'Anna and Fontana. It is shown that if R is generically Cohen-Macaulay (resp. generically Gorenstein) and I is generically maximal Cohen-Macaulay (resp. generically canonical module), then $R{\bowtie}I$ is generically Cohen-Macaulay (resp. generically Gorenstein). We also de ned generically quasi-Gorenstein ring and we investigate when $R{\bowtie}I$ is generically quasi-Gorenstein. In addition, it is shown that $R{\bowtie}I$ is approximately Cohen-Macaulay if and only if R is approximately Cohen-Macaulay, provided some special conditions. Finally it is shown that if R is approximately Gorenstein, then $R{\bowtie}I$ is approximately Gorenstein.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.4
• /
• pp.645-652
• /
• 2002

In this note we investigate some results which concern the types of local rings. In particular it is shown that if the type of a quasi-unmixed local ring A is less than or equal to depth A ＋ 1, and $\hat{A}_p$ is Cohen-Macaulay for every prime $p\neq\hat{m}$, then A is Cohen-Macaulay. (This implies the previously known result: if A satisfies $(S_{n-1})}$, where n is the type of a .ins A, then A is Cohen-Macaulay.)

• Journal of the Korean Mathematical Society
• /
• v.60 no.3
• /
• pp.683-694
• /
• 2023

Let a be an ideal of 𝔞 commutative noetherian ring R. We give some descriptions of the 𝔞-depth of 𝔞-relative Cohen-Macaulay modules by cohomological dimensions, and study how relative Cohen-Macaulayness behaves under flat extensions. As applications, the perseverance of relative Cohen-Macaulayness in a polynomial ring, formal power series ring and completion are given.

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

• Journal of Integrative Natural Science
• /
• v.11 no.2
• /
• pp.82-86
• /
• 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
• /
• v.51 no.2
• /
• pp.373-386
• /
• 2014

Let (R,m) be a commutative Noetherian local ring. In this paper we show that a finitely generated R-module M of dimension d is Cohen-Macaulay if and only if there exists a proper ideal I of R such that depth($M/I^nM$) = d for $n{\gg}0$. Also we show that, if dim(R) = d and $I_1{\subset}\;{\cdots}\;{\subset}I_n$ is a chain of ideals of R such that $R/I_k$ is maximal Cohen-Macaulay for all k, then $n{\leq}{\ell}_R(R/(a_1,{\ldots},a_d)R)$ for every system of parameters $a1,{\ldots},a_d$ of R. Also, in the case where dim(R) = 2, we prove that the ideal transform $D_m(R/p)$ is minimax balanced big Cohen-Macaulay, for every $p{\in}Assh_R$(R), and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.

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

• Communications of the Korean Mathematical Society
• /
• v.11 no.4
• /
• pp.887-893
• /
• 1996

Let (R,m) be a Cohen-Macaulay local ring with an infinite residue field and let $J = (a_1, \cdots, a_l)$ be a minimal reduction of an equimultiple ideal I of R. In this paper we shall prove that the following conditions are equivalent: (1) $I^2 = JI$. (2) $gr_I(R)/mgr_I(R)$ is Cohen-Macaulay with minimal multiplicity at its maximal homogeneous ideal N. (3) $N^2 = (a'_1, \cdots, a'_l)N$, where $a'_i$ denotes the images of $a_i$ in I/mI for $i = 1, \cdots, l$.