For a ring endomorphism of a ring R, Krempa called

rigid endomorphism if

= 0 implies a = 0 for a

R, and Hong et al. called R an

-rigid ring if there exists a rigid endomorphism

. 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.,

-Armendariz rings and

-skew Armendariz rings) by Hong et al. In this paper, we study the relationship between

-rigid rings and extended Armendariz rings, and so we get various conditions on the rings which are equivalent to the condition of being an

-rigid ring. Several known results relating to extended Armendariz rings can be obtained as corollaries of our results.