Rings whose singular ideals are nil

Abstract

It is well known that when a ringRsatisfies ACC on right annihilators of elements, then the right singular ideal ofRis nil, in this case, we sayRis right nil-singular. Many classes of rings whose singular ideals are nil, but do not satisfy the ACC on right annihilators, are presented and the behavior of them is investigated with respect to various constructions, in particular skew polynomial rings and triangular matrix rings. The class of right nil-singular rings contains pi-regular rings and is closed under direct sums. Examples are provided to explain and delimit our results.