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In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let $R$ be a commutative ring with nonzero identity and $Q$ a proper ideal of $R$. Then $Q$ is called strongly quasi primary if $ab\in Q$ for $a,b\in R$ implies either $a^{2}\in Q$ or $b^{n}\in Q~ (a^{n}\in Q$ or $b^{2}\in Q)$ for some $n\in \mathbb{N} $. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph $\Gamma_{I}(R)$ and denote it by $\Gamma_{I}^{\ast}(R)$, where $I$ is an ideal of $R$. We investigate the relations between $\Gamma_{I}^{\ast} (R)$ and $\Gamma_{I}(R)$. Further, we use strongly quasi primary ideals and $\Gamma_{I}^{\ast}(R)$ to characterize von Neumann regular rings.
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참고문헌 (19)
참고문헌 신청
M. Achraf / 2017 / 2-absorbing ideals in formal power series rings / Palestine J. Math 6(2):502~506
D. D. Anderson / 1994 / Ideals generated by powers of elements / Bull. Australian Math. Soc 49(3):373~376
D. D. Anderson / 2009 / Idealization of a module / J. Commutative Algebra 1(1):3~56
D. F. Anderson / 2003 / Zero-divisor graphs, von Neumann regular rings, and Boolean algebras / J. Pure Appl. Algebra 180(3):221~241
D. F. Anderson / 1999 / The zero-divisor graph of a commutative ring / J. Algebra 217(2):434~447
D. D. Anderson / 1994 / Ideals generated by powers of elements / Bull. Australian Math. Soc 49(3):373~376
D. D. Anderson / 2009 / Idealization of a module / J. Commutative Algebra 1(1):3~56
D. F. Anderson / 2003 / Zero-divisor graphs, von Neumann regular rings, and Boolean algebras / J. Pure Appl. Algebra 180(3):221~241
D. F. Anderson / 1999 / The zero-divisor graph of a commutative ring / J. Algebra 217(2):434~447
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