NCI环的多项式环未必是NCI环  

Polynomial rings over NCI rings need not be NCI rings

作　　者：陈卫星[1] 

机构地区：[1]山东工商学院数学与信息科学学院,山东烟台264005

出　　处：《辽宁师范大学学报（自然科学版）》2008年第2期135-137,共3页Journal of Liaoning Normal University:Natural Science Edition

基　　金：山东省教育厅自然科学基金资助项目(J07YH03)

摘　　要：环R指的是结合环但未必含有单位元.环R称为NCI环如果当它的幂零元集合N(R)≠0时那么N(R)包含R的一个非零理想.主要研究有关NCI环的性质,证明了存在NCI环R但是R的多项式环R[x]非NCI环,这否定地回答了S.U.Hwang等人(Bull.Korean Math.Soc.44(2007),No.2)的一个公开问题.进一步证明了如果环R的多项式环R[x]是NCI环则R是NCI环.此外还证明了存在NCI环但它的幂级数环不是NCI环,而如果环R的幂级数环为R[[x]]是NCI环那么R是NCI环.最后证明了如果环R存在非零的局部幂零理想I那么R的全矩阵环Mn(R)是NCI环.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.

关 键 词：NCI环 诣零根 多项式环 幂级数环 

分 类 号：O153.3[理学—数学]