On bicommutators of modules over \(H\)-separable extension rings. III (Q1334347)

This is the third paper of the author on the study of the properties of the bicommutator of modules [for part II cf. ibid. 20, No. 3, 601-608 (1991; Zbl 0749.16019)]. Here he improves some of the theorems in the first paper [ibid. 20, No. 3, 497-501 (1991; Zbl 0749.16018)] by looking at extensions that generalize the bicommutator extension situation. The author also studies the Galois condition of the extension of bicommutators of a left \(A\)-module \(M\). The idea is to translate the properties of the \(H\)-separable extension \(A\) over \(B\) to the corresponding extension \(A^*\) over \(B^*\), where \(A^*\) is the bicommutator of \(A\) in the \(A\)-module \(M\) (this is the endomorphism ring of \(M\) as a left module over the endomorphism ring of \(M\) as a left \(A\)-module), and \(B^*\) likewise. When the \(H\)-separable extension \(A\) over \(B\) has the extra condition that \(A\) is a left f.g. projective \(B\)-module, the extension \(A^*\) over \(B^*\) is also \(H\)-separable, as proved by the author in part I. This situation is assumed for the main results of this paper. First, the author shows that if \(B\) also satisfies the double centralizer condition in \(A\), then \(_AM\) has the double centralizer condition if and only if \(_BM\) has the double centralizer condition (or \(A\cong A^*\) iff \(B\cong B^*\)). As a result of the proof, the case when \(A\) is a right f.g. projective \(B\)-module also holds. Second, the author gives a one-to-one correspondence between the subrings of \(A^*\) containing \(A\) and the subrings of \(B^*\) containing \(B\) which commute with \(A\). This correspondence also preserves localizations. Third, the author shows that if the \(H\)-separable extension \(A\) over \(B\) is also a \(G\)-Galois extension, then the same occurs for the extension \(A^*\) over \(B^*\). The technique used is to prove the statements for some general extensions \(S\) over \(T\) and then apply them to the case \(S=A^*\) and \(T=B^*\). The general situation is as follows. The ring \(A\) is an extension of the subring \(B\) with the same unity, \(S\) is an extension of \(A\), \(T\) is a subring of \(S\) and an extension of \(B\). The main property of this structure, which is the hypothesis of all the theorems and propositions, is the ``centralizer property'', that is the centralizer of \(A\) in \(S\) is the center of \(S\), the centralizer of \(B\) in \(S\) coincides with the centralizer of \(T\) in \(S\), and \(T\) satisfies the double centralizer property in \(S\). In particular, the bicommutator extension \(A^*\) over \(B^*\) satisfies this property. The author also gives an example of these structures which are not bicommutators; \(A\), \(B\) are valuation rings, \(S\) is the completion of \(A\) and \(T\) is the closure of \(B\).