Maximal non-nonnil-principal ideal rings

Abstract

Let H be the set of all commutative rings with unity whose nilradical is a divided prime ideal. The concept of maximal non-nonnil-PIR is introduced to generalize the concept of maximal non-PID. A ring extension R⊂T in H is a called a maximal non-nonnil-principal ideal ring if R is not a nonnil-principal ideal ring but each subring of T properly containing R is a nonnil-principal ideal ring. It is shown that R+XT[X] (respectively, R+XT[[X]]) is a maximal non-nonnil-PIR subring of T[X] (respectively, T[[X]]) if and only if R+XT[X] (respectively, R+XT[[X]]) is a maximal non-PID subring of T[X] (respectively, T[[X]]).