We consider some properties of Armendariz and rigid rings. We prove that the direct product of rigid (weak rigid), weak Armendariz rings is a rigid (weak rigid), weak Armendariz ring. On the assumption that the factor ring $R/I$ is weak Armendariz, where $I$ is nilpotent ideal, we prove that $R$ is a weak Armendariz ring. We also prove that every ring isomorphism preserves weak skew Armendariz structure. Armendariz rings of Laurent power series are also considered.