∗-ARMENDARIZ PROPERTY FOR INVOLUTION RINGS

Abstract

In this paper we study Armendariz property for ∗-rings. We introduce the class of ∗-Armendariz ∗-rings, which contains reduced ∗-rings, and
its properties are studied. We prove that each ∗-Armendariz ∗-ring is ∗- Abelian. Moreover, we show that the property of a ∗-Armendariz ∗-ringR is extended to its polynomial ∗-ring R[x], localization S−1R of R to S, Laurent polynomial ∗-ring R[x, x−1] and from Ore ∗-ring to its classical Quotient Q. Furthermore, we prove that for a ∗-Armendariz ∗-ring R; R is ∗-Baer if and only if R[x] (resp., R[[x]]) is also ∗-Baer. Finally, we show that the property of ∗-ring having quasi-∗-IFP R can be extendeded to its localization of R to S, Laurent polynomial ∗-ring and polynomial ∗-ring.