Direct sum decompositions of conformal rings
The main purpose of this paper is to discuss decomposition theorems for certain classes of rings. The prototype for the decompositions discussed appears in a paper by \textit{A. W. Chatters} and the reviewer [J. Lond. Math. Soc., II. Ser. 33, 22-32 (1986; Zbl 0601.16001)], which discusses Noetherian UFRs, these being prime Noetherian rings in which every non- zero prime ideal contains a non-zero prime ideal generated by a normal element. It is shown there that if \(R\) is a Noetherian ring in which every non-minimal prime ideal contains a non-zero prime ideal which is principal on both sides, then \(R\) decomposes as a direct sum of Noetherian UFRs and rings with no non-minimal prime ideals. The rings for which the author considers the possibilities for such results are termed conformal rings. The reviewer [Glasg. Math. J. 31, 103-113 (1989; Zbl 0672.16001)] called a ring conformal if every non-zero ideal contains a non-zero normal element and considered only prime conformal rings. In order to consider non-prime rings, the author here defines a ring \(R\) with prime radical \(P\) to be conformal if every ideal of \(R\) which is essential modulo \(P\), as a bisubmodule of \(R\), contains a normal element \(n\) such that \(P\subseteq nR\). A decomposition theorem is proved for conformal rings which are either Noetherian (on both sides) or have the property that every ideal is generated by a finite normalising sequence. Other aspects of the paper are discussions of alternative definitions of conformality and of the role of conditions on two-sided ideals, a comprehensive list of examples, and another decomposition theorem generalising that given by Chatters and the reviewer.
scientific article; zbMATH DE number 575879
scientific article