On Strongly Quasi Primary Ideals
Bull. Korean Math. Soc.
Published online 2019 May 16
SUAT KOC, UNSAL TEKIR, and GULSEN ULUCAK
Marmara University, Gebze Technical University
Abstract : In this paper, we introduce strongly quasi primary ideals which is an intermediate classes 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\in Q\ $or $b^{n}\in Q\ (a^{n}\in Q\ $or $b\in Q)\ $for some $n\in%
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.\ $ 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. In section 3, we construct a subgraph of ideal based zero divisor graph $\Gamma_{I}(R)$ and denote by $\Gamma_{I}^{\ast}(R),\ $where $I\ $is an ideal of $R.\ $We investigate the relations between $\Gamma_{I}^{\ast}%(R)\ $and $\Gamma_{I}(R)$. We give a characterization of von Neumann regular rings in terms of strongly quasi primary ideals and $\Gamma_{I}^{\ast}(R).$
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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