• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.251-258
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• 2002

Let R be a commutative Noetherian ring and M an R-module. In this paper we will consider the asymptotic behaviour of ideals relative to an R-module E which is M-injective.

• Journal of applied mathematics & informatics
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• v.39 no.3_4
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• pp.259-266
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• 2021

Let R be a commutative ring with identity and let S be a nonempty subset of R. We define S to be an almost multiplicative subset of R if for each a, b ∈ S, there exist integers m, n ≥ 1 such that a^mbⁿ ∈ S. In this article, we study some utilization of almost multiplicative subsets.

• 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 the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.799-809
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• 2013

An ideal I of a local ring (R, m, k) is said to be m-full if there exists an element $x{\in}m$ such that Im : x = I. An ideal I of a local ring R is said to have the Rees property if ${\mu}$(I) > ${\mu}$(J) for any ideal J containing I. We study properties of m-full ideals and we characterize m-primary m-full ideals in terms of the minimal number of generators of the ideals. In particular, for a m-primary ideal I of a 2-dimensional regular local ring (R, m, k), we will show that the following conditions are equivalent. 1. I is m-full 2. I has the Rees property 3. ${\mu}$(I)=o(I)+1 In this paper, let (R, m, k) be a commutative Noetherian local ring with infinite residue field k = R/m.

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

• The Pure and Applied Mathematics
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• v.7 no.1
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• pp.33-40
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• 2000

We prove some characterization of rings with chain conditions in terms of fuzzy quotient rings and fuzzy ideals. We also show that a ring R is left Artinian if and only of the set of values of every fuzzy ideal on R is upper well-ordered.

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

• Bulletin of the Korean Mathematical Society
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• v.30 no.1
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• pp.71-77
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• 1993

Every ring considered in the paper will be assumed to be commutative and have a unit element. An ideal A of a ring R will be called primal if the elements of R which are zero divisors modulo A, form an ideal of R, say pp. If A is a primal ideal of R, P is called the adjoint ideal of A. The adjoint ideal of a primal ideal is prime [2]. The definition of primal ideals may also be formulated as follows: An ideal A of a ring R is primal if in the residue class ring R/A the zero divisors form an ideal of R/A. If Q is a primary idel of a ring R then every zero divisor of R/Q is nilpotent; therefore, Q is a primal ideal of R. That a primal ideal need not be primary, is shown by an example in [2]. Let R[X], and R[[X]] denote the polynomial ring and formal power series ring in an indeterminate X over a ring R, respectively. Let S be a multiplicative system in a ring R and S$^{-1}$ R the quotient ring of R. Let Q be a P-primary ideal of a ring R. Then Q[X] is a P[X]-primary ideal of R[X], and S$^{-1}$ Q is a S$^{-1}$ P-primary ideal of a ring S$^{-1}$ R if S.cap.P=.phi., and Q[[X]] is a P[[X]]-primary ideal of R[[X]] if R is Noetherian [1]. We search for analogous results when primary ideals are replaced with primal ideals. To show an ideal A of a ring R to be primal, it sufficies to show that a-b is a zero divisor modulo A whenever a and b are zero divisors modulo A.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1225-1236
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• 2016

Let R be a commutative ring with $1{\neq}0$ and n a positive integer. In this article, we introduce the n-Krull dimension of R, denoted $dim_n\;R$, which is the supremum of the lengths of chains of n-absorbing ideals of R. We study the n-Krull dimension in several classes of commutative rings. For example, the n-Krull dimension of an Artinian ring is finite for every positive integer n. In particular, if R is an Artinian ring with k maximal ideals and l(R) is the length of a composition series for R, then $dim_n\;R=l(R)-k$ for some positive integer n. It is proved that a Noetherian domain R is a Dedekind domain if and only if $dim_n\;R=n$ for every positive integer n if and only if $dim_2\;R=2$. It is shown that Krull's (Generalized) Principal Ideal Theorem does not hold in general when prime ideals are replaced by n-absorbing ideals for some n > 1.

• The Pure and Applied Mathematics
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• v.1 no.1
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• pp.1-6
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• 1994

In this paper, some properties for a PI-ring satisfying the descending chain condition on essential left ideals are studied: Let R be a ring with a polynomial identity satisfying the descending chain condition on essential ideals. Then all minimal prime ideals in R are maximal ideals. Moreover, if R has only finitely many minimal prime ideals, then R is left and right Artinian. Consequently, if every primeideal of R is finitely generated as a left ideal, then R is left and right Artinian. A finitely generated PI-algebra over a commutative Noetherian ring satisfying the descending chain condition on essential left ideals is a finite module over its center.(omitted)