On the 2-Absorbing Ideals in Commutative Rings

Sh. Payrovi ; S. Babaei

Bulletin of the Malaysian Mathematical Society, Tome 36 (2013) no. 4

Résumé

Let $R$ be a commutative ring with identity. In this article, we study a generalization of prime ideal. A proper ideal $I$ of $R$ is called a 2-absorbing ideal if whenever $abc\in I$ for $a,b,c\in R$, then $ab\in I$ or $bc\in I$ or $ac\in I$. It is shown that if $I$ is a 2-absorbing ideal of a Noetherian ring $R$, then $R/I$ has some ideals $J_n$, where $1\leq n\leq t$ and $t$ is a positive integer, such that $J_n$ possesses a prime filtration $F_{J_n}:\ \ 0\subset R(x_1+I)\subset \bigotimes R(x_1+I)\ R(x_2+I)\subset\cdots \subset R(x_1+I) \bigotimes\ \cdots\ \bigotimes R(x_n+I)=J_n$ with $\ Ass_R(J_n)=\{ I:_Rx_i\ \ |\ \ i=1,... ,n\}$ and $|\ Ass_R(J_n)|=n$. Also, a 2-Absorbing Avoidance Theorem is proved

Détail

Classification : 13A15