Abstract
Let R be a maximal order and A, B be R-ideals of R. Clearly ( A B ) ∗ ⊇ B ∗ A ∗ is satisfied and if R is a Dedekind prime ring, the equality holds, i.e., ( A B ) ∗ = 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 ( A B ) ∗ = 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 [ X t ] associated to an invertible ideal X are also PI G-Dedekind prime rings.
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