On generalized Dedekind prime rings.
type: Item
instance of: scholarly article
title: On generalized Dedekind prime rings.
zbMATH Open document ID: 1171.16001
published in: Journal of Algebra
publication date: 2008-12-02T00:00:00Z
review text: \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 \(R\).
reviewed by: D. D. Anderson
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16U10
zbMATH DE Number: 5376938
zbMATH Keywords: generalized Dedekind prime rings
zbMATH Keywords: invertible ideals
zbMATH Keywords: reflexive ideals
zbMATH Keywords: maximal orders
zbMATH Keywords: integral domains
author: E. Akalan
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.07.001
OpenAlex ID: W2094386098
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DOI: 10.1016/J.JALGEBRA.2008.07.001
description: scientific article; zbMATH DE number 5376938