On the 2-absorbing ideals in commutative rings (Q384128)

\textit{A. Badawi} [Bull. Aust. Math. Soc. 75, No. 3, 417--429 (2007; Zbl 1120.13004)] generalized the concept of prime ideals. He defined a nonzero proper ideal \(I\) of \(R\) to be a \(2\)-absorbing ideal of \(R\) if whenever \(a, b, c\in R\) and \(abc\in I\), then \(ab\in I\) or \(ac\in I\) or \(bc\in I\). In the paper under review, the authors among the other results prove that 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 fitration \(\displaystyle F_{J_n} : 0\subset R(x_1 +I)\subset R(x_1 +I)\oplus R(x_2 +I)\subset \cdots \subset R(x_1 +I)\oplus \cdots \oplus R(x_n +I) = J_n\) with \(\text{Ass}_R(J_n) = \{ I :_R x_i | i = 1, \cdots, n\}\) and \(|\text{Ass}_R(J_n)| = n\). They also prove the \(2\)-Absorbing Avoidance Theorem for ideals as a generalization of the Prime Avoidance Theorem.