• Bulletin of the Korean Mathematical Society
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• v.56 no.3
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• pp.729-743
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• 2019

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 ${\Gamma}^*_I(R)$, where I is an ideal of R. We investigate the relations between ${\Gamma}^*_I(R)$ and ${\Gamma}_I(R)$. Further, we use strongly quasi primary ideals and ${\Gamma}^*_I(R)$ to characterize von Neumann regular rings.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.93-104
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• 2024

Let R be a commutative ring with identity. In this paper, we introduce a new class of ideals called the class of strongly quasi J-ideals lying properly between the class of J-ideals and the class of quasi J-ideals. A proper ideal I of R is called a strongly quasi J-ideal if, whenever a, b ∈ R and ab ∈ I, then a² ∈ I or b ∈ Jac(R). Firstly, we investigate some basic properties of strongly quasi J-ideals. Hence, we give the necessary and sufficient conditions for a ring R to contain a strongly quasi J-ideals. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the primary ideals, the prime ideals and the maximal ideals. Finally, we give an idea about some strongly quasi J-ideals of the quotient rings, the localization of rings, the polynomial rings and the trivial rings extensions.