Strongly irreducible ideals of a commutative ring (Q5957268)

Let \(R\) be a commutative ring with identity. The authors define an ideal \(I\) of \(R\) to be strongly irreducible if for ideals \(J\) and \(K\) of \(R\), the inclusion \(J\cap K\subseteq I\) implies that \(J\subseteq I\) or \(K\subseteq I\). Clearly a prime ideal is strongly irreducible and a strongly irreducible ideal is irreducible (i.e., it is not the intersection of two ideals that properly contain it).NEWLINENEWLINENEWLINEA number of results and examples are given. Perhaps the main theorem is that for a non-prime ideal \(I\) of height greater than zero in a Noetherian ring \(R\), \(I\) is strongly irreducible if and only if \(I\) is primary, \(R_P\) is a DVR, where \(P=\text{rad}(I)\), and \(I=P^n\) for some integer \(n>1\).