Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON STRONGLY QUASI PRIMARY IDEALS

  • Koc, Suat (Department of Mathematics Marmara University) ;
  • Tekir, Unsal (Department of Mathematics Marmara University) ;
  • Ulucak, Gulsen (Department of Mathematics Faculty of Science Gebze Technical University)
  • Received : 2018.05.31
  • Accepted : 2019.02.07
  • Published : 2019.05.31
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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 ${\Gamma}^*_I(R)$, 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.

Keywords

E1BMAX_2019_v56n3_729_f0001.png 이미지

Figure 1. Relations between strongly quasi primary ideals and other classical ideals.

E1BMAX_2019_v56n3_729_f0002.png 이미지

Figure 3. $\Gamma^{\ast}_{0}(\mathbb{Z}_{12})$ vs $\Gamma_{0}(\mathbb{Z}_{12})$

E1BMAX_2019_v56n3_729_f0003.png 이미지

Figure 2. $\Gamma^{\ast}_{0}(\mathbb{Z}_6)$

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