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RISS 인기검색어
- 검색키워드
- 저자 : Masoome Zahiri (검색결과 5 건)
- 무료
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ON ANNIHILATIONS OF IDEALS IN SKEW MONOID RINGS
Mohammadi, Rasul,Moussavi, Ahmad,Zahiri, Masoome Korean Mathematical Society 2016 대한수학회지 Vol.53 No.2
According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and ${\sigma}:M{\rightarrow}End(R)$ a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and ${\sigma}:M{\rightarrow}End(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).
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A note on minimal prime ideals
Rasul Mohammadi,Ahmad Moussavi,Masoome Zahiri 대한수학회 2017 대한수학회보 Vol.54 No.4
Let $R$ be a strongly $2$-primal ring and $I$ a proper ideal of $R$. Then there are only finitely many prime ideals minimal over $I$ if and only if for every prime ideal $P$ minimal over $I$, the ideal $P/\sqrt{I}$ of $R/\sqrt{I}$ is finitely generated if and only if the ring $R/\sqrt{I}$ satisfies the \emph{ACC} on right annihilators. This result extends ``D. D. Anderson, A note on minimal prime ideals, \emph{Proc. Amer. Math. Soc.} 122 (1994), no. 1, 13--14." to large classes of noncommutative rings. It is also shown that, a $2$-primal ring $R$ only has finitely many minimal prime ideals if each minimal prime ideal of $R$ is finitely generated. Examples are provided to illustrate our results.
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On annihilations of ideals in skew monoid rings
Rasul Mohammadi,Ahmad Moussavi,Masoome Zahiri 대한수학회 2016 대한수학회지 Vol.53 No.2
According to Jacobson \cite{Jacobson}, a right ideal is bounded if it contains a non-zero ideal, and Faith \cite{Faith2} called a ring strongly right bounded if every non-zero right ideal is bounded. From \cite{Hwang}, a ring is strongly right $AB$ if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property ($A$) and the conditions asked by Nielsen \cite{Nielsen}. It is shown that for a u.p.-monoid $M$ and $\sigma: M \rightarrow {\rm End}(R)$ a compatible monoid homomorphism, if $R$ is reversible, then the skew monoid ring $R\ast M$ is strongly right $AB$. If $R$ is a strongly right $AB$ ring, $M$ is a u.p.-monoid and $\sigma: M \rightarrow {\rm End}(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring $R\ast M$ has right Property $(A)$.
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A NOTE ON MINIMAL PRIME IDEALS
Mohammadi, Rasul,Moussavi, Ahmad,Zahiri, Masoome Korean Mathematical Society 2017 대한수학회보 Vol.54 No.4
Let R be a strongly 2-primal ring and I a proper ideal of R. Then there are only finitely many prime ideals minimal over I if and only if for every prime ideal P minimal over I, the ideal $P/{\sqrt{I}}$ of $R/{\sqrt{I}}$ is finitely generated if and only if the ring $R/{\sqrt{I}}$ satisfies the ACC on right annihilators. This result extends "D. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math. Soc. 122 (1994), no. 1, 13-14." to large classes of noncommutative rings. It is also shown that, a 2-primal ring R only has finitely many minimal prime ideals if each minimal prime ideal of R is finitely generated. Examples are provided to illustrate our results.
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Rasul Mohammadi,Ahmad Moussavi,Masoome Zahiri Korean Mathematical Society 2023 대한수학회지 Vol.60 No.6
We introduce two subclasses of abelian McCoy rings, so-called π-CN-rings and π-duo rings, and systematically study their fundamental characteristic properties accomplished with relationships among certain classical sorts of rings such as 2-primal rings, bounded rings etc. It is shown that a ring R is π-CN whenever every nilpotent element of index 2 in R is central. These rings naturally generalize the long-known class of CN-rings, introduced by Drazin [9]. It is proved that π-CN-rings are abelian, McCoy and 2-primal. We also show that, π-duo rings are strongly McCoy and abelian and also they are strongly right AB. If R is π-duo, then R[x] has property (A). If R is π-duo and it is either right weakly continuous or every prime ideal of R is maximal, then R has property (A). A π-duo ring R is left perfect if and only if R contains no infinite set of orthogonal idempotents and every left R-module has a maximal submodule. Our achieved results substantially improve many existing results.
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