On generalized Dedekind prime rings

Let R be a maximal order and A, B be R-ideals of R. Clearly ∗(AB)⊇B∗A∗ is satisfied and if R is a Dedekind prime ring, the equality holds, i.e., ∗(AB)=B∗A∗. However, the equality is not true in general. In this paper, we answer the question: If R is a maximal order when is ∗(AB)=B∗A∗ for all non-zero R-ideals of R? We call prime Noetherian maximal orders satisfying this property, generalized Dedekind prime rings. We give several characterizations of G-Dedekind prime rings and show that being a G-Dedekind prime ring is a Morita invariant. Moreover, we prove that if R is a PI G-Dedekind prime ring then the polynomial ring R[x] and the Rees ring R[Xt] associated to an invertible ideal X are also PI G-Dedekind prime rings.