http://chineseinput.net/에서 pinyin(병음)방식으로 중국어를 변환할 수 있습니다.
변환된 중국어를 복사하여 사용하시면 됩니다.
Mohamed Chhiti,Soibri Moindze 대한수학회 2023 대한수학회지 Vol.60 No.2
Let $R$ be a commutative ring with identity and $S$ be a multiplicatively closed subset of $R$. In this article we introduce a new class of ring, called $S$-multiplication rings which are $S$-versions of multiplication rings. An $R$-module $M$ is said to be $S$-multiplication if for each submodule $N$ of $M$, $sN\subseteq JM\subseteq N$ for some $s\in S$ and ideal $J$ of $R$ (see for instance \cite[Definition 1]{DA.TA.UTSK}). An ideal $I$ of $R$ is called $S$-multiplication if $I$ is an $S$-multiplication $R$-module. A commutative ring $R$ is called an $S$-multiplication ring if each ideal of $R$ is $S$-multiplication. We characterize some special rings such as multiplication rings, almost multiplication rings, arithmetical ring, and $S$-$PIR$. Moreover, we generalize some properties of multiplication rings to $S$-multiplication rings and we study the transfer of this notion to various context of commutative ring extensions such as trivial ring extensions and amalgamated algebras along an ideal.
B\'ezout rings and weakly B\'ezout rings
Haitham El Alaoui 대한수학회 2022 대한수학회보 Vol.59 No.4
In this paper, we study some properties of B\'ezout and weakly B\'ezout rings. Then, we investigate the transfer of these notions to trivial ring extensions and amalgamated algebras along an ideal. Also, in the context of domains we show that the amalgamated is a B{\'e}zout ring if and only if it is a weakly B\'ezout ring. All along the paper, we put the new results to enrich the current literature with new families of examples of non-B\'ezout weakly B\'ezout rings.
ALMOST WEAKLY FINITE CONDUCTOR RINGS AND WEAKLY FINITE CONDUCTOR RINGS
Choulli, Hanan,Alaoui, Haitham El,Mouanis, Hakima Korean Mathematical Society 2022 대한수학회논문집 Vol.37 No.2
Let R be a commutative ring with identity. We call the ring R to be an almost weakly finite conductor if for any two elements a and b in R, there exists a positive integer n such that a<sup>n</sup>R ∩ b<sup>n</sup>R is finitely generated. In this article, we give some conditions for the trivial ring extensions and the amalgamated algebras to be almost weakly finite conductor rings. We investigate the transfer of these properties to trivial ring extensions and amalgamation of rings. Our results generate examples which enrich the current literature with new families of examples of nonfinite conductor weakly finite conductor rings.
ZPI Property In Amalgamated Duplication Ring
Achraf Malek,Ahmed Hamed 경북대학교 자연과학대학 수학과 2022 Kyungpook mathematical journal Vol.62 No.2
Let A be a commutative ring. We say that A is a ZPI ring if every proper ideal of A is a finite product of prime ideals [5]. In this paper, we study when the amalgamated duplication of A along an ideal I, A I to be a ZPI ring. We show that if I is an idempotent ideal of A, then A is a ZPI ring if and only if A I is a ZPI ring.
El Maalmi, Mourad,Mouanis, Hakima Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.2
An element in a ring R with identity is called invo-clean if it is the sum of an idempotent and an involution and R is called invoclean if every element of R is invo-clean. Let C(R) be the center of a ring R and g(x) be a fixed polynomial in C(R)[x]. We introduce the new notion of g(x)-invo clean. R is called g(x)-invo if every element in R is a sum of an involution and a root of g(x). In this paper, we investigate many properties and examples of g(x)-invo clean rings. Moreover, we characterize invo-clean as g(x)-invo clean rings where g(x) = (x-a)(x-b), a, b ∈ C(R) and b - a ∈ Inv(R). Finally, some classes of g(x)-invo clean rings are discussed.
S-COHERENT PROPERTY IN TRIVIAL EXTENSION AND IN AMALGAMATED DUPLICATION
Mohamed Chhiti,Salah Eddine Mahdou Korean Mathematical Society 2023 대한수학회논문집 Vol.38 No.3
Bennis and El Hajoui have defined a (commutative unital) ring R to be S-coherent if each finitely generated ideal of R is a S-finitely presented R-module. Any coherent ring is an S-coherent ring. Several examples of S-coherent rings that are not coherent rings are obtained as byproducts of our study of the transfer of the S-coherent property to trivial ring extensions and amalgamated duplications.
When every finitely generated regular ideal is finitely presented
Mohamed Chhiti,Salah Eddine Mahdou 대한수학회 2024 대한수학회논문집 Vol.39 No.2
In this paper, we introduce a weak version of coherent that we call regular coherent property. A ring is called regular coherent, if every finitely generated regular ideal is finitely presented. We investigate the stability of this property under localization and homomorphic image, and its transfer to various contexts of constructions such as trivial ring extensions, pullbacks and amalgamated. Our results generate examples which enrich the current literature with new and original families of rings that satisfy this property.
ON ALMOST QUASI-COHERENT RINGS AND ALMOST VON NEUMANN RINGS
El Alaoui, Haitham,El Maalmi, Mourad,Mouanis, Hakima Korean Mathematical Society 2022 대한수학회보 Vol.59 No.5
Let R be a commutative ring with identity. We call the ring R to be an almost quasi-coherent ring if for any finite set of elements α<sub>1</sub>, …, α<sub>p</sub> and a of R, there exists a positive integer m such that the ideals $\bigcap{_{i=1}^{p}}\;R{\alpha}^m_i$ and Ann<sub>R</sub>(α<sup>m</sup>) are finitely generated, and to be almost von Neumann regular rings if for any two elements a and b in R, there exists a positive integer n such that the ideal (α<sup>n</sup>, b<sup>n</sup>) is generated by an idempotent element. This paper establishes necessary and sufficient conditions for the Nagata's idealization and the amalgamated algebra to inherit these notions. Our results allow us to construct original examples of rings satisfying the above-mentioned properties.
AMALGAMATED MODULES ALONG AN IDEAL
El Khalfaoui, Rachida,Mahdou, Najib,Sahandi, Parviz,Shirmohammadi, Nematollah Korean Mathematical Society 2021 대한수학회논문집 Vol.36 No.1
Let R and S be two commutative rings, J be an ideal of S and f : R → S be a ring homomorphism. The amalgamation of R and S along J with respect to f, denoted by R ⋈f J, is the special subring of R × S defined by R ⋈f J = {(a, f(a) + j) | a ∈ R, j ∈ J}. In this paper, we study some basic properties of a special kind of R ⋈f J-modules, called the amalgamation of M and N along J with respect to , and defined by M ⋈ JN := {(m, (m) + n) | m ∈ M and n ∈ JN}, where : M → N is an R-module homomorphism. The new results generalize some known results on the amalgamation of rings and the duplication of a module along an ideal.
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<sub>Nil(R)</sub> 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<sub>Nil(R)</sub> and 𝜙 restricted to R is also a ring homomorphism from R into R<sub>Nil(R)</sub> 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<sub>i</sub>, where each R<sub>i</sub> 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<sub>i</sub>. We show that the theories of 𝜙-pseudo Krull rings resemble those of pseudo-Krull domains.