On n-Absorbing Ideals and Two Generalizations of Semiprime Ideals

Hojjat Mostafanasab, Ahmad Yousefian Darani

Abstract

Let R be a commutative ring and n be a positive integer. A proper ideal I of R is called an n-absorbing ideal if whenever x_1...x_{n+1}\in I for x_1,...,x_{n+1}\in R, then there are n of the x_i's whose product is in I. We give a generalization of Prime Avoidance Theorem. Also, an n-Absorbing Avoidance Theorem is proved. Moreover, we introduce the notions of quasi-n-absorbing ideals and of semi-n-absorbing ideals. We say that a proper ideal I of R is a quasi-n-absorbing ideal if whenever a^nb\in I for a,b\in R, then a^n\in I or a^{n-1}b\in I. A proper ideal I of R is said to be a semi-n-absorbing ideal if whenever a^{n+1}\in I for a\in R, then a^n\in I.