Weak Bourbaki unmixed rings: a step towards non-noetherian Cohen-Macaulayness.
The aim of this paper is to provide an analogue to the Cohen-Macaulay property for the case of non-noetherian rings. The author considers the weak Bourbaki unmixedness property as being one of the possible candidates. In detail, a commutative ring \(R\) is said to be weak Bourbaki unmixed if for every ideal \(I\) that can be generated by at most \(\roman{height}(I)\) elements, its minimal weak associated primes (in the sense of Bourbaki) are not embedded. Theorem 1 proves that this notion coincides with the classical one of Cohen-Macaulayness if \(R\) is noetherian, and moreover it satisfies some transfer properties. For instance, it is proved that if the polynomial ring \(R[X]\) is weak Bourbaki unmixed, then so is \(R\), and if every localization \(R_{\mathfrak{p}}\) at a prime \(\mathfrak{p}\) is weak Bourbaki unmixed, then so is \(R\). At the end the author shows that a polynomial ring \(R[X_1,X_2,\dots]\) over a Cohen-Macaulay ring \(R\) is weak Bourbaki unmixed, and gives several low-dimensional examples of rings that are (not) weak Bourbaki unmixed.
scientific article; zbMATH DE number 2144263