Bull. Korean Math. Soc. Published online July 23, 2019

R. Nikandish, M. J. Nikmehr, and A. Yassine
Department of Mathematics, Jundi-Shapur University of Technology, Dezful, Iran, Faculty of Mathematics, K.N. Toosi University of Technology

Abstract : 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