• 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.

• 대한수학회논문집
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• 제39권3호
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• pp.623-642
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• 2024

This paper introduces and studies a generalization of (n, d)-rings introduced and studied by Costa in 1994 to rings with prime nilradical. Among other things, we establish that the ϕ-von Neumann regular rings are exactly either ϕ-(0, 0) or ϕ-(1, 0) rings and that the ϕ-Prüfer rings which are strongly ϕ-rings are the ϕ-(1, 1) rings. We then introduce a new class of rings generalizing the class of n-coherent rings to characterize the nonnil-coherent rings introduced and studied by Bacem and Benhissi.

• 대한수학회논문집
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• 제32권3호
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• pp.543-552
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• 2017

Let R be a commutative ring with identity, and ${\phi}:{\mathfrak{I}}(R){\rightarrow}{\mathfrak{I}}(R){\cup}\{{\varnothing}\}$ be a function where ${\mathfrak{I}}(R)$ is the set of all ideals of R. Following [2], a proper ideal P of R is called a ${\phi}$-prime ideal if $x,y{\in}R$ with $xy{\in}P-{\phi}(P)$ implies $x{\in}P$ or $y{\in}P$. For an ideal I of R, we define the ${\phi}$-radical ${\sqrt[{\phi}]{I}}$ to be the intersection of all ${\phi}$-prime ideals of R containing I, and show that this notion inherits most of the essential properties of the usual notion of radical of an ideal. We also investigate when the set of all ${\phi}$-prime ideals of R, denoted $Spec_{\phi}(R)$, has a Zariski topology analogous to that of the prime spectrum Spec(R), and show that this topological space is Noetherian if and only if ${\phi}$-radical ideals of R satisfy the ascending chain condition.

• 대한수학회보
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• 제44권2호
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• pp.225-231
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• 2007

Given an injective envelope E of a left R-module M, there is an associative Galois group Gal$({\phi})$. Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope $E[x^{-1}]$ of an inverse polynomial module $M[x^{-1}]$ as a left R[x]-module and we can define an associative Galois group Gal$({\phi}[x^{-1}])$. In this paper we describe the relations between Gal$({\phi})$ and Gal$({\phi}[x^{-1}])$. Then we extend the Galois group of inverse polynomial module and can get Gal$({\phi}[x^{-s}])$, where S is a submonoid of $\mathbb{N}$ (the set of all natural numbers).

• 대한수학회보
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• 제53권5호
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• pp.1341-1352
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• 2016

Let R be a commutative Noetherian ring, ${\Phi}$ a system of ideals of R and $I{\in}{\Phi}$. In this paper among other things we prove that if M is finitely generated and $t{\in}\mathbb{N}$ such that the R-module $H^i_{\Phi}(M)$ is $FD_{{\leq}1}$ (or weakly Laskerian) for all i < t, then $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all i < t and for any $FD_{{\leq}0}$ (or minimax) submodule N of $H^t_{\Phi}(M)$, the R-modules $Hom_R(R/I,H^t_{\Phi}(M)/N)$ and $Ext^1_R(R/I,H^t_{\Phi}(M)/N)$ are finitely generated. Also it is shown that if cd I = 1 or $dimM/IM{\leq}1$ (e.g., $dim\;R/I{\leq}1$) for all $I{\in}{\Phi}$, then the local cohomology module $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all $i{\geq}0$. These generalize the main results of Aghapournahr and Bahmanpour [2], Bahmanpour and Naghipour [6, 7]. Also we study cominimaxness and weakly cofiniteness of local cohomology modules with respect to a system of ideals.

• East Asian mathematical journal
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• 제24권2호
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• pp.139-144
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• 2008

Given an injective envelope E of a left R-module M, there is an associative Galois group Gal($\phi$). Let R be a left noetherian ring and E be an injective envelope of M, then there is an injective envelope E[$x^{-1}$] of an inverse polynomial module M[$x^{-1}$] as a left R[x]-module and we can define an associative Galois group Gal(${\phi}[x^{-1}]$). In this paper we extend the Galois group of inverse polynomial module and can get Gal(${\phi}[x^{-s}]$), where S is a submonoid of $\mathds{N}$ (the set of all natural numbers).

• 대한수학회보
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• 제30권1호
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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.