Muhittin Ba\c{s}er, Fatma Kaynarca, and Tai Keun Kwak Kocatepe University, Kocatepe University, Daejin University

Abstract : For a ring endomorphism $\alpha$ of a ring $R$, Krempa called $\alpha$ a rigid endomorphism if $a\alpha(a)=0$ implies $a=0$ for $a\in R$, and Hong et al. called $R$ an $\alpha$-rigid ring if there exists a rigid endomorphism $\alpha$. 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., $\alpha$-Armendariz rings and $\alpha$-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between $\alpha$-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an $\alpha$-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.