Direct sum decompositions of conformal rings.

*(English)*Zbl 0808.16004The 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.

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.

Reviewer: D.A.Jordan (Sheffield)

##### 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 |

##### Keywords:

decomposition theorems; prime Noetherian rings; normal element; direct sum of Noetherian UFRs; conformal rings; prime radical; normalising sequence; two-sided ideals
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\textit{E. A. Whelan}, Acta Math. Hung. 62, No. 3--4, 211--230 (1993; Zbl 0808.16004)

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##### References:

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