For a ring endomorphism α, we introduce weak α-rigid and weak α-skew Armendariz rings which are a generalization of α-rigid rings, and investigated their properties. Moreover, we prove that a ring R is weak α-rigid if and only if for any n, the n by n upper triangular matrix ring Tn(R) is weak α-rigid. If R is semicommutative and weak α-rigid, it is proven that the ring R[x] and the ring (R[x]/(xn)) where (xn) is the ideal generated by xn and n is a positive integer, are weak α-skew Armendariz.