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Direct sum decompositions of conformal rings. (English) Zbl 0808.16004
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 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.
MSC:
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16U30 Divisibility, noncommutative UFDs
16P40 Noetherian rings and modules (associative rings and algebras)
16D20 Bimodules in associative algebras
16D25 Ideals in associative algebras
16N60 Prime and semiprime associative rings
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