Abstarct: We observe from known results that the set of nilpotent elements in Armendariz rings has an important role. The upper nilradical coincides with the prime radical in Armendariz rings. So it can be shown that the factor ring of an Armendariz ring over its prime radical is also Armendariz, with the help of Antoine’s results for nil-Armendariz rings. We study the structure of rings with such property in Armendariz rings and introduce APR as a generalization. It is shown that APR is placed between Armendariz and nil-Armendariz. It is shown that an APR ring which is not Armendariz, can always be constructed from any Armendariz ring. It is also proved that a ring R is APR if and only if so is R[x] and that N(R[x]) = N(R)[x] when R is APR, where R[x] is the polynomial ring with an indeterminate x over R and N(R) denotes the set of all nilpotent elements. Several kinds of APR rings are found or constructed in the precess related to ordinary ring constructions.