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Extending the notion of a reduced ring, we call a right module M over a ring R a reduced module if, for any m Є M and a€ R, ma= 0 implies mR Ma= 0. Various results of reduced rings are extended to reduced modules. Moreover, these modules are used to obtain results on certain annihilator conditions of the polynomial extension and the power series extension of a module.

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 [[x]] 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") is an Armendariz ring where an Armendariz ring is any ring S such that if (ax)(objx³)= 0 in S [x] then a, b,= 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 Є 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 [[x]]. A ring R is called a Baer (resp. right pp-) 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 pp) iff so is R [x], and there exist non-reduced Baer rings whose polynomial ring is not