Rigidness and Extended Armendariz Property

Abstract

Abstract. For a ring endomorphism α of a ring R, Krempa called α a rigid endomorphism if aα(a) = 0 implies a = 0 for a ∈ R, and Hong et al. called R an α-rigid ring if there exists a rigid endomorphism α. Due to Rege and Chhawchharia, a ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. The Armendariz property of polynomials was extended to one of skew polynomials (i.e., α-Armendariz rings and α-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between α-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an α-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.Throughout this paper, all rings are associative with identity. Given a ring R, the polynomial ring over R is denoted by R [x]. Recall that a ring R is called reduced if it has no nonzero nilpotent elements. Armendariz [1, Lemma 1] showed that for a reduced ring R, if any polynomial [2,3,5,6,8,10,11,12].The reducedness and Armendariz property of a ring were extended as follows. For a ring R with a ring endomorphism α : R → R, a skew polynomial ring (also called an Ore extension of endomorphism type) R[x; α] of R is the ring obtained by giving the polynomial ring over R with the new multiplication xr = α(r)x for all r ∈ R. Recall that an endomorphism α of a ring R is called rigid [9] if aα(a) = 0 implies a = 0 for a ∈ R, and a ring R is called α-rigid [4] if there exists a rigid endomorphism α of R. Note that any rigid endomorphism of a ring is a monomorphism, and α-rigid rings are reduced rings [4, Proposition 5]. On the other hand, the Armendariz property with respect to polynomials was extended to one of skew polynomials. A ring R is called α-Armendariz (resp.,