Bull. Korean Math. Soc. 2020; 57(1): 117-126
Online first article July 23, 2019 Printed January 31, 2020
https://doi.org/10.4134/BKMS.b190094
Copyright © The Korean Mathematical Society.
Reza Nikandish, Mohammad Javad Nikmehr, Ali Yassine
Jundi-Shapur University of Technology; K.N. Toosi University of Technology; K.N. Toosi University of Technology
Let $R$ be a commutative ring with identity. A proper ideal $I$ of $R$ is called $2$-prime if for all $a,b\in R$ such that $ab\in I$, then either $a^2$ or $b^2$ lies in $I$. In this paper, we study $2$-prime ideals which are generalization of prime ideals. Our study provides an analogous to the prime avoidance theorem and some applications of this theorem. Also, it is shown that if $R$ is a PID, then the families of primary ideals and $2$-prime ideals of $R$ are identical. Moreover, a number of examples concerning $2$-prime ideals are given. Finally, rings in which every $2$-prime ideal is a prime ideal are investigated.
Keywords: $2$-prime ideal, $2$-prime avoidance theorem, $2$-$P$ ring
MSC numbers: Primary 13A15, 13C05
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