About composite rings (Q2785950)

Let \(A\) be a subring of a ring \(B\). The composite ring of the pair \((A,B)\) is \(R: =A+ XB[X]\). The paper is devoted to a study of composite rings, which are very good tools for (counter)examples. Browsing through the given results, one can choose some of them as follows.NEWLINENEWLINENEWLINEA factor ring of a composite ring \(R\) is the idealization of \(B\) in \(A\) and there is a natural surjective morphism from the symmetric algebra \(S_A(B)\) to \(R\). Flatness criteria for composite ring morphisms are given. If \(I\) is the ideal \(XB [X]\) of \(R\), properties of \(I\) and \(R\) are closely related. For example, \(I\) is finitely generated if and only if \(A\to B\) is finite. This enables the author to prove that \(R\) is noetherian (resp. locally noetherian) if and only if \(A\to B\) is finite (resp. locally finite) and \(A\) and \(B\) are noetherian (resp. locally noetherian). An example of a locally noetherian, non-noetherian ring is thus obtained. Arithmetical properties of composite rings are also studied using normal pairs. The seminormalization of a composite ring is computed. Finally, if \(A\to B\) is flat and finite, it is shown that \(R\) is Cohen-Macaulay if and only if \(A\) and \(B\) are Cohen-Macaulay.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].