Armendariz rings and gaussian rings

摘要

We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ringRis (Gaussian) Armendariz if for two polynomialsf,g#x2208;RX (the ideal ofRgenerated by the coefficients off gis the product of the ideals generated by the coefficients offandg)fg= 0 impliesaibj=0 for each coefficientaioffandbjofg. A number of examples of Armendariz rings are given. We show thatRArmendariz implies thatRX is Armendariz and that forRvon Neumann regularRis Armendariz if and only ifRis reduced. We show thatRis Gaussian if and only if each homomorphic image ofRis Armendariz. Characterizations of whenRX andRX are Gaussian are given.