More on the 2-prime ideals of commutative rings
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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.
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.
참고문헌 (8)
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- R. Y. Sharp, "Steps in Commutative Algebra" (2000), Cambridge University Press