• 호남수학학술지
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• 제29권3호
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• pp.351-366
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• 2007

Let R be a commutative ring and let M be an R-module. Let X = Spec(M) be the prime spectrum of M with Zariski topology. Our main purpose in this paper is to specify the topological dimensions of X, where X is a Noetherian topological space, and compare them with those of topological dimensions of $Supp_{R}$(M). Also we will give a characterization for the irreducibility of X and we obtain some related results.

• 대한수학회지
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• 제60권2호
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• pp.407-464
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• 2023

Let R be a commutative ring with identity. The structure theorem says that R is a PIR (resp., UFR, general ZPI-ring, π-ring) if and only if R is a finite direct product of PIDs (resp., UFDs, Dedekind domains, π-domains) and special primary rings. All of these four types of integral domains are Krull domains, so motivated by the structure theorem, we study the prime factorization of ideals in a ring that is a finite direct product of Krull domains and special primary rings. Such a ring will be called a general Krull ring. It is known that Krull domains can be characterized by the star operations v or t as follows: An integral domain R is a Krull domain if and only if every nonzero proper principal ideal of R can be written as a finite v- or t-product of prime ideals. However, this is not true for general Krull rings. In this paper, we introduce a new star operation u on R, so that R is a general Krull ring if and only if every proper principal ideal of R can be written as a finite u-product of prime ideals. We also study several ring-theoretic properties of general Krull rings including Kaplansky-type theorem, Mori-Nagata theorem, Nagata rings, and Noetherian property.

• 대한수학회논문집
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• 제35권4호
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• pp.1095-1106
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• 2020

The purpose of this paper is to introduce a new class of rings that is closely related to the class of pseudo-Krull domains. Let 𝓗 = {R | R is a commutative ring and Nil(R) is a divided prime ideal of R}. Let R ∈ 𝓗 be a ring with total quotient ring T(R) and define 𝜙 : T(R) → R_Nil(R) by ${\phi}({\frac{a}{b}})={\frac{a}{b}}$ for any a ∈ R and any regular element b of R. Then 𝜙 is a ring homomorphism from T(R) into R_Nil(R) and 𝜙 restricted to R is also a ring homomorphism from R into R_Nil(R) given by ${\phi}(x)={\frac{x}{1}}$ for every x ∈ R. We say that R is a 𝜙-pseudo-Krull ring if 𝜙(R) = ∩ R_i, where each R_i is a nonnil-Noetherian 𝜙-pseudo valuation overring of 𝜙(R) and for every non-nilpotent element x ∈ R, 𝜙(x) is a unit in all but finitely many R_i. We show that the theories of 𝜙-pseudo Krull rings resemble those of pseudo-Krull domains.

• 대한수학회보
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• 제34권1호
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• pp.35-42
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• 1997

Throughout this paper, all rings are assumed to be commutative with identity. By a local ring (R,m), we mean a Noetherian ring R which has a unique maximal ideal m. Let I be an ideal in a ring R and t an indeterminate over R.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제11권1호
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• pp.23-27
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• 2004

We calculate dim $\hat{A}$ which is a completion of a Noetherian ring A with respect to I-adic topology. We do not use localization but power series techniques.

• 대한수학회보
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• 제59권6호
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• pp.1349-1357
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• 2022

Let 𝒜 = (A_n)_n≥0 be an increasing sequence of rings. We say that 𝓘 = (I_n)_n≥0 is an associated sequence of ideals of 𝒜 if I₀ = A₀ and for each n ≥ 1, I_n is an ideal of A_n contained in I_n+1. We define the polynomial ring and the power series ring as follows: $I[X]\, = \,\{\, f \,=\, {\sum}_{i=0}^{n}a_iX^i\,{\in}\,A[X]\,:\,n\,{\in}\,\mathbb{N},\,a_i\,{\in}\,I_i \,\}$ and $I[[X]]\, = \,\{\, f \,=\, {\sum}_{i=0}^{+{\infty}}a_iX^i\,{\in}\,A[[X]]\,:\,a_i\,{\in}\,I_i \,\}$. In this paper we study the Noetherian and the SFT properties of these rings and their consequences.

• 대한수학회논문집
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• 제36권2호
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• pp.197-207
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• 2021

The main purpose of this paper is to study the zero-divisor properties of the zero-symmetric near-ring of skew formal power series R₀[[x; α]], where R is a symmetric, α-compatible and right Noetherian ring. It is shown that if R is reduced, then the set of all zero-divisor elements of R₀[[x; α]] forms an ideal of R₀[[x; α]] if and only if Z(R) is an ideal of R. Also, if R is a non-reduced ring and annR(a - b) ∩ Nil(R) ≠ 0 for each a, b ∈ Z(R), then Z(R₀[[x; α]]) is an ideal of R₀[[x; α]]. Moreover, if R is a non-reduced right Noetherian ring and Z(R₀[[x; α]]) forms an ideal, then ann_R(a - b) ∩ Nil(R) ≠ 0 for each a, b ∈ Z(R). Also, it is proved that the only possible diameters of the zero-divisor graph of R₀[[x; α]] is 2 and 3.

• 호남수학학술지
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• 제3권1호
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• pp.109-113
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• 1981

본(本) 논문(論文)에서는 Noetherian Ring 상(上)의 finitely generated injective module이 completely indecomposable modules의 direct sum으로 표시(表示)될 필요충분조건(必要充分條件)을 구(求)하였다.

• 대한수학회논문집
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• 제9권2호
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• pp.261-264
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• 1994

Throughout this paper, all rings are assumed to be commutative with identity. By a local ring (R, m), we mean a Noetherian ring R which has the unique maximal ideal m. By dim(R) we always mean the Krull dimension of R. Let I be an ideal in a ring R and t an indeterminate over R. Then the Rees algebra R[It] is defined to be(omitted)

• 대한수학회보
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• 제57권5호
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• pp.1165-1176
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• 2020

Let M be a module over a commutative ring R. In this paper, we study Int(R, M), the module of integer-valued polynomials on M over R, and Int_M(R), the ring of integer-valued polynomials on R over M. We establish some properties of Krull dimensions of Int(R, M) and Int_M(R). We also determine when Int(R, M) and Int_M(R) are nontrivial. Among the other results, it is shown that Int(ℤ, M) is not Noetherian module over Int_M(ℤ) ∩ Int(ℤ), where M is a finitely generated ℤ-module.