For a ring endomorphism of a ring R, Krempa called $alpha$ 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.