Armendariz rings and Gaussian rings

We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ring R is (Gaussian) Armendariz if for two polynomials f, g is an element of R[X] (the ideal of R generated by the coefficients of fg is the product of the ideals generated by the coefficients of f and g) fg = 0 implies a(i)b(j) = 0 for each coefficient a(i) of f and b(j) of g. A number of examples of Armendariz rings are given. We show that R Armendariz implies that R[X] is Armendariz and that for R von Neumann regular, R is Armendariz if and only if R is reduced. We show that R is Gaussian if and only if each homomorphic image of R is Armendariz. Characterizations of when R[X] and R[X] are Gaussian are given.