Bulletin of the
Korean Mathematical Society BKMS

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