On strongly quasi primary ideals
Bull. Korean Math. Soc. 2019 Vol. 56, No. 3, 729-743
https://doi.org/10.4134/BKMS.b180522
Published online May 31, 2019
Suat Koc, Unsal Tekir, Gulsen Ulucak
Marmara University; Marmara University; Gebze Technical University
Abstract : 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.
Keywords : valuation domain, divided ring, strongly quasi primary ideal, zero divisor graph, ideal based zero divisor graph
MSC numbers : 13F30, 13A15, 05C25.
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