Polynomial rings over NCI rings need not be NCI rings

Rings R are associative but not necessarily have identities. A ring R is called a NCI ring if when the set of its power zero elements N(R)≠0 N(R) contains a nonzero ideal of R. We mainly study the properties of NCI rings. It is proved that the polynomial ring over a NCI ring needs not be a NCI ring, negatively answering a question from S. U. Hwang et al. (Bull. Korean Math. Soc. 44 (2007), No.2). Furthermore, it is proved that if the polynomial ring is a NCI ring then so is R. Also it is proved that there exists a ring over which the power series ring is not a NCI ring and that if it is a NCI ring then so is R. Finally it is shown that if R is a ring with a nonzero local powerzero ideal then the matrix ring is a NCI ring.