On generalized Dedekind prime rings.
\textit{D. D. Anderson} and \textit{B. G. Kang} [J. Algebra 122, No. 2, 323-336 (1989; Zbl 0698.13004)] and \textit{M. Zafrullah} [Mathematika 33, 285-295 (1986; Zbl 0613.13001)] investigated integral domains \(R\) satisfying \((AB)^{-1}=A^{-1} B^{-1}\) for all nonzero fractional ideals \(A\) and \(B\) of \(R\). This paper extends these studies to the noncommutative case. Let \(R\) be a maximal order and \(A\) and \(B\) be \(R\)-ideals of \(R\). So clearly \((AB)^*\supseteq B^*A^*\) and we have equality if \(R\) is a Dedekind prime ring. It is easily seen that a maximal order satisfies the equality \((AB)^*=B^*A^*\) if and only if every reflexive \(R\)-ideal is invertible. The author defines a generalized Dedekind prime ring \(R\) (G-Dedekind prime ring) to be a prime Noetherian maximal order with every reflexive \(R\)-ideal invertible. A number of equivalent conditions for \(R\) being a G-Dedekind prime ring are given and it is shown that being a G-Dedekind prime ring is a Morita invariant. Finally, it is shown that if \(R\) is a PI G-Dedekind prime ring, then so is the polynomial ring \(R[X]\) and the Rees ring \(R[IX]\) where \(I\) is an invertible ideal of \(R\).
scientific article; zbMATH DE number 5376938
scientific article