RINGS OVER WHICH POLYNOMIAL RINGS ARE ARMENDARIZ AND REVERSIBLE

Abstract. A ring R is called reversibly Armendariz if bjai = 0 for all i; j whenever f(x)g(x) = 0 for two polynomials f(x) =∑m i=0 aix i; g(x) = ∑n j=0 bjx j over R. It is proved that a ring R is reversibly Armendariz if and only if its polynomial ring is re-versibly Armendariz if and only if its Laurent polynomial ring is re-versibly Armendariz. Relations between reversibly Armendariz rings and related ring properties are examined in this note, observing the structures of many examples concerned. Various kinds of reversibly Armendariz rings are provided in the process. Especially it is shown to be possible to construct reversibly Armendariz rings from given any Armendariz rings. 1. Reversibly Armendariz rings Throughout this section every ring is associative (possibly without identity). Let R be a ring. The polynomial ring with an indeterminate x over R is denoted by R[x]. Cf(x) denotes the set of all coefficients of f(x) ∈ R[x]. The n by n full (resp. upper triangular) matrix ring over R is denoted by Matn(R) (resp. Un(R)). Use eij for the matrix with (i; j)-entry 1 and elsewhere 0. Let Z (Zn) denotes the ring of integers (modulo n). A ring is usually called reduced if it has no nonzero nilpotent elements. Following Rege and Chhawchharia [19], a ring R is called Armendari