Trace properties in rings with zero divisors
Let \(D\) be an integral domain with quotient field \(K\) and let for a nonzero ideal \(I\) of \(D,\) \(I^{-1}=D:I=\{x\in K:xI\subseteq D\}\). \textit{D. D. Anderson, J. A. Huckaba} and \textit{I. J. Papick} [Houston J. Math. 13, 13--17 (1987; Zbl 0624.13002)] showed that if \(D\) is a valuation domain then for every nonzero ideal \(I\) of \(D,\) either \(II^{-1}=D \) or \(II^{-1}\) is a prime ideal of \(D.\) Fontana, Huckaba and Papick, realizing \(II^{-1}\) as the trace of the \(D\)-module \(I\), introduced ``domains with trace property'' or TP-domains as domains \(D\) such that for each nonzero ideal \(I\) of \(D,\) \(II^{-1}=D\) or \(II^{-1}\) is a prime ideal of \(D.\) They looked for TP-domains among Noetherian and Prufer domains. Later, \textit{W. J. Heinzer} and \textit{I. J. Papick} [J. Algebra 112, No. 1, 110--121 (1988; Zbl 0641.13001)] extended the playing field \ by defining domains with the radical trace property (RTP) as domains \(D\) such that for each nonzero non-invertible ideal \(I\) of \(D\) the ideal \(II^{-1}\) is a radical ideal\(.\) A number of algebraists, including the first author, S. Kabbaj and A. Mimouni have worked on topics related to this RTP-property. The ideal \(II^{-1}=I(D:I)\) is also known as the trace ideal. Often when a topic has been sufficiently explored for integral domains some researchers turn their attention to seeing if similar results can be obtained for rings with zero-divisors. I tend to think that this trend started, in earnest, in the sixties and grew with \textit{J. Huckaba} [Commutative rings with zero divisors. New York etc.: Marcel Dekker, Inc. (1988; Zbl 0637.13001)], who was the advisor of the first author of the paper under review. In any case it did produce some interesting results and the paper under view is one of them. So, from now on, let \(R\) be a commutative ring with \(1\neq 0\), \(T(R)\) the total quotient ring of \(R,\) \(Q(R)\) the complete ring of quotients and let \(Q_{0}(R)\) be the ring of finite fractions over \(R.\) Each of these ``quotient rings'' can be defined in terms of equivalence classes of \(R\)-module homomorphisms on certain ideals of \(R\). The nonzero ideals get replaced by dense ideals \(I\) (such that \(\mathrm{ann}(I)=(0))\) or by semiregular (\(I\) that contains a finitely generated dense ideal) or by regular (\(I\) that contains a nonzero divisor or regular element). With these replacements for ``nonzero ideal'' the authors show that if \(I\) is a dense ideal of \(R\) then \(I(R:I)\) is the trace of the \(R\)-module \(I.\) Then they study rings for each of whose regular (semiregular) non-invertible ideals \(I,\) \(I(R:I)\) is a radical ideal and call them rings with radical trace RTP (resp., \(Q_{0}\)-RTP) property. Let's call \(S\) a \(T(R)\) (resp., \(Q_{0}(R))\) overring of \(R\) if \(S\) is a subring of \(T(R)\) ( resp., \(Q_{0}(R)\)) and \(S\) contains \(R\) as a subring. The authors prove that if \(S\) is a \(T(R)\) overring of \(R\) and if \(S\) is \(R\) -flat then \(S\) is RTP whenever \(R\) is. On the other hand if \(S\) is a flat \( Q_{0}(R)\)-overring of \(R\) and \(R\) is \(Q_{0}\)-RTP then so is \(S.\)
scientific article; zbMATH DE number 6005466
scientific article