Some remarks on $S$-valuation domains

Ali Benhissi,Abdelamir Dabbabi 대한수학회 2024 대한수학회논문집 Vol.39 No.1

Let $A$ be a commutative integral domain with identity element and $S$ a multiplicatively closed subset of $A$. In this paper, we introduce the concept of $S$-valuation domains as follows. The ring $A$ is said to be an $S$-valuation domain if for every two ideals $I$ and $J$ of $A$, there exists $s\in S$ such that either $sI\subseteq J$ or $sJ\subseteq I$. We investigate some basic properties of $S$-valuation domains. Many examples and counterexamples are provided.

FINITELY t-VALUATIVE DOMAINS

Chang, Gyu Whan The Kangwon-Kyungki Mathematical Society 2014 한국수학논문집 Vol.22 No.4

Let D be an integral domain with quotient field K. In [1], the authors called D a finitely valuative domain if, for each $0{\neq}u{\in}K$, there is a saturated chain of rings $D=D_0{\varsubsetneq}D_1{\varsubsetneq}{\cdots}{\subseteq}$ $D_n=D[x]$, where x = u or $u^{-1}$. They then studied some properties of finitely valuative domains. For example, they showed that the integral closure of a finitely valuative domain is a Pr$\ddot{u}$fer domain. In this paper, we introduce the notion of finitely t-valuative domains, which is the t-operation analog of finitely valuative domains, and we then generalize some properties of finitely valuative domains.

ON ALMOST PSEUDO-VALUATION DOMAINS

Chang, Gyu Whan The Kangwon-Kyungki Mathematical Society 2010 한국수학논문집 Vol.18 No.2

Let D be an integral domain, and let ${\bar{D}}$ be the integral closure of D. We show that if D is an almost pseudo-valuation domain (APVD), then D is a quasi-$Pr{\ddot{u}}fer$ domain if and only if D=P is a quasi-$Pr{\ddot{u}}fer$ domain for each prime ideal P of D, if and only if ${\bar{D}}$ is a valuation domain. We also show that D(X), the Nagata ring of D, is a locally APVD if and only if D is a locally APVD and ${\bar{D}}$ is a $Pr{\ddot{u}}fer$ domain.

Directed unions of local quadratic transforms of a regular local ring

Heinzer, W.,Kim, M.K.,Toeniskoetter, M. Academic Press 2017 Journal of algebra Vol.488 No.-

<P>Let (R, m) be a d-dimensional regular local domain with d >= 2 and let V be a valuation domain birationally dominating R such that the residue field of V is algebraic over R/m. Let v be a valuation associated to V. Associated to R and there exists an infinite directed family {(R-n,m(n))}n >= 0 of d- dimensional regular local rings dominated by V with R = R-0 and Rn+1 the local quadratic transform of R-n along V. Let S := boolean OR(n >= 0) R-n. Abhyankar proves that S = V if d = 2. Shannon oBserves that often S is properly contained in V if d >= 3, and Granja gives necessary and sufficient conditions for S to be equal to V. The directed family {(R-n,m(n))}(n >= 0) and the integral domain S = boolean OR(n >= 0) R-n may be defined without first prescribing a dominating valuation domain V. If {{(R-n,m(n))}(n >= 0) switches strongly infinitely often, then S = V is a rank one valuation domain and for nonzero elements f and g in m, we have v(f)/v(g) = lim(n ->infinity)ordR(n)(f)/ordR(n)(g). If {(R-n,m(n))}(n >= 0) is a family of monomial local quadratic transforms, we give necessary and sufficient conditions for {(R-n,m(n))}(n >= 0) to switch strongly infinitely often. If these conditions hold, then S = V is a rank one valuation domain of rational rank d and v is a monomial valuation. Assume that V is rank one and birationally dominates S. Let s = Sigma(infinity)(i=0) v(m(i)) Granja, Martinez and Rodriguez show that s = infinity implies S = V. We prove that s is finite if V has rational rank at least 2. In the case where V has maximal rational rank, we give a sharp upper bound for s and show that s attains this bound if and only if the sequence switches strongly infinitely often. (C) 2017 Elsevier Inc. All rights reserved.</P>

Locally pseudo-valuation domains with only finitely many star operations

Houston, E.,Lee, E.K.,Park, M.H. Academic Press 2015 Journal of algebra Vol.444 No.-

Let R be a locally pseudo-valuation domain. For each maximal ideal M of R, denote by V(M) the associated valuation domain of the pseudo-valuation domain R<SUB>M</SUB>, and let T=@?V(M). We characterize those R that have only finitely many star operations. When (R:T)≠(0), the characterization becomes: R has only finitely many star operations if and only if T and each R<SUB>M</SUB> have only finitely many star operations. On the other hand, if (R:T)=(0), then neither direction of this equivalence holds, and we give examples to illustrate the various possibilities.

PRIMARY DECOMPOSITION OF SUBMODULES OF A FREE MODULE OF FINITE RANK OVER A BÉZOUT DOMAIN

Fatemeh Mirzaei,Reza Nekooei 대한수학회 2023 대한수학회보 Vol.60 No.2

Let $R$ be a commutative ring with identity. In this paper, we characterize the prime submodules of a free $R$-module $F$ of finite rank with at most $n$ generators, when $R$ is a $\text{GCD}$ domain. Also, we show that if $R$ is a B\'ezout domain, then every prime submodule with $n$ generators is the row space of a prime matrix. Finally, we study the existence of primary decomposition of a submodule of $F$ over a B\'ezout domain and characterize the minimal primary decomposition of this submodule.

Prime t-ideals in power series rings over a discrete valuation domain

Academic Press 2010 Journal of algebra Vol.324 No.12

Let V be an m-dimensional discrete valuation domain. It is known that the power series ring V@?x@? has t-dimension m. We will show that V@?x<SUB>1</SUB>,...,x<SUB>n</SUB>@? has t-dimension 2m-1 for all n>=2.

Normal Pairs of Going-down Rings

Dobbs, David Earl,Shapiro, Jay Allen Department of Mathematics 2011 Kyungpook mathematical journal Vol.51 No.1

Let (R, T) be a normal pair of commutative rings (i.e., R ${\subseteq}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R ${\subseteq}$ S ${\subseteq}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.

ON 𝜙-PSEUDO-KRULL RINGS

El Khalfi, Abdelhaq,Kim, Hwankoo,Mahdou, Najib Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.4

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.

On $\phi$-pseudo-Krull rings

Abdelhaq El Khalfi,김환구,Najib Mahdou 대한수학회 2020 대한수학회논문집 Vol.35 No.4

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