SKEW LAURENT POLYNOMIAL EXTENSIONS OF BAER AND P.P.-RINGS

Abstract

Let R be a ring and fi a monomorphism of R. We study the skew Laurent polynomial rings R(x,x i1 ;fi) over an fi-skew Armendariz ring R. We show that, if R is an fi-skew Armendariz ring, then R is a Baer (resp. p.p.-)ring if and only if R(x,x i1 ;fi) is a Baer (resp. p.p.-) ring. Consequently, if R is an Armendariz ring, then R is a Baer (resp. p.p.-)ring if and only if R(x,xi1) is a Baer (resp. p.p.-)ring. Throughout this paper R denotes an associative ring with unity and fi : R ! R is an endomorphism, which is not assumed to be surjective. We denote R(x;fi) the Ore extension whose elements are the polynomials § n=0rix i , ri 2 R, where the addition is defined as usual and the multiplication subject to the relation xa = fi(a)x for any a 2 R. The set {x j }j‚0 is easily seen to be a left Ore subset of R(x ;fi), so that one can localize R(x;fi) and form the skew Laurent polynomial ring R(x,x i1 ;fi). Elements of R(x,x i1 ;fi) are finite sums of elements of the form x ij rx i , where r 2 R and i and j are nonnegative integers. A ring R is called Armendariz if whenever polynomials f(x) = a0+a1x+···+ anx n , g(x) = b0+b1x+···+bmx m 2 R(x) satisfy f(x)g(x) = 0, then aibj = 0 for each i,j. The term Armendariz was introduced by Rege and Chhawchharia (20). This nomenclature was used by them since it was Armendariz (2, Lemma 1) who initially showed that a reduced ring (i.e., a ring without nonzero nilpotent elements) always satisfies this condition. According to Krempa (16), an endomorphism fi of a ring R is called to be rigid if afi(a) = 0 implies a = 0 for a 2 R. A ring R is called fi-rigid if there exists a rigid endomorphism fi of R. Note that any rigid endomorphism of a ring is a monomorphism and fi-rigid rings are reduced rings by Hong et al. (10).