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RISS 인기검색어
- 검색키워드
- 저자 : Hosein Fazaeli Moghimi (검색결과 4 건)
- 무료
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- 유료
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ON n-ABSORBING IDEALS AND THE n-KRULL DIMENSION OF A COMMUTATIVE RING
Moghimi, Hosein Fazaeli,Naghani, Sadegh Rahimi Korean Mathematical Society 2016 대한수학회지 Vol.53 No.6
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.
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On $n$-absorbing ideals and the $n$-Krull dimension of a commutative ring
Hosein Fazaeli Moghimi,Sadegh Rahimi Naghani 대한수학회 2016 대한수학회지 Vol.53 No.6
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 $\dim_nR=n$ for every positive integer $n$ if and only if $\dim_2R=2$. 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 $n>1$.
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GENERALIZED QUASI-PRIMARY RINGS
Hosein Fazaeli Moghimi,MAHDI SAMIEI 장전수학회 2018 Advanced Studies in Contemporary Mathematics Vol.28 No.4
In this paper, the structure of commutative rings with identity all of whose ideals are quasi-primary, called generalized quasi-primary rings, is studied and several equivalent conditions to such rings are considered. Equivalently, a generalized quasi-primary ring may be viewed as a ring whose the set of radical ideals forms a chain. It is proved that an Artinian local ring R is a generalized quasi-primary ring and the converse is true if R is a non-domain Noetherian ring.
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A GENERALIZATION OF THE PRIME RADICAL OF IDEALS IN COMMUTATIVE RINGS
Harehdashti, Javad Bagheri,Moghimi, Hosein Fazaeli Korean Mathematical Society 2017 대한수학회논문집 Vol.32 No.3
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.
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