On the primitivity of polynomial rings with nonprimitive coefficient rings

Abstract:

For a hereditary noetherian prime ring $R$ with classical quotient ring $Q$, various necessary and sufficient conditions are given for the polynomial ring $R[{X_1}, \ldots ,{X_n}]$ to be primitive when $R$ itself is not primitive. It is shown that if $R$ is a local hereditary noetherian prime ring, then $R[X]$ is primitive if and only if $Q[X]$ is primitive. Similarly, for a semilocal hereditary noetherian prime ring $R$ whose Jacobson radical contains a nonzero central element, it is shown that $R[{X_1}, \ldots ,{X_n}]$ is primitive if and only if $Q[{X_1}, \ldots ,{X_n}]$ is primitive.