ABSTRACT
All rings R are associative and have identity, and modules are unitary right modules. A ring R is called a reduced ring if a2 = 0 in R always implies a = 0. The notion of reduced rings has been studied by many authors. Some of the known results on reduced rings can be recalled as follows: R is reduced iff R[x] is reduced iff -R[[a;]] is reduced; R is reduced iff R is a subdirect product of domains by Andrunakievic and Rjabuhin [2]; recently it was proved in Anderson and Camillo [1] that, for n > 2, R is reduced iff R [ x ] / ( x n ) is an Armendariz ring where an Armendariz ring is any ring 5 such that if (53iLoa»a;I)(I3j=o^'a;J') = 0 in S[x] then a,ibj — 0 for all i and j. For an endomorphism a of R, Krempa [12] obtained that the skew polynomial ring R[x; a] is reduced iff R is a-rigid, that is, for any a € R, aa(a) = 0 implies a = 0. The concept of a reduced ring is very useful in the investigation of certain annihilator conditions of R[x] and .R[[#]]. A ring R is called a Baer (resp. right p.p.-) ring if the right annihilator of any non-empty subset (resp. any element) of R is generated by an idempotent of R. A well-known result of Armendariz [3] states that, for a reduced ring R, R is Baer (resp. right p.p.) iff so is R[x], and there exist non-reduced Baer rings whose polynomial ring is not
Baer. Recently, this result has been extended in several directions by BirkenmeierKim-Park [5], Han-Hirano-Kim [6], Hirano [7], Hong-Kim-Kwak [8], and Kim-Lee [11].
