On bicommutators of modules over \(H\)-separable extension rings (Q1185403)

The main result of this paper is as follows. Let \(A\), \(B\) be rings such that \(A\) is an \(H\)-separable extension of \(B\) and is finitely generated projective as a left \(B\)-module. Let \(M\) be a left \(A\)-module, let \(A^*\) denote the bicommutator \(\text{Bic}(_ AM)=\text{End}(_ \Delta M)\), \(\Delta=\text{End}(_ AM)\), \(B^*=\text{Bic}(_ BM)\). Then \(A^*\) is an \(H\)-separable extension of \(B^*\), \(A^*\) is finitely generated projective over \(B^*\) and \(B^*=V_{A^*}(V_{A^*}(B^*))\) where \(V_{A^*}(S)\) is the centralizer of \(S\) in \(A^*\) for any \(S\subseteq A^*\). If, as well, \(B\) is a left \(B\)-direct summand of \(A\), then \(A=A^*\) if and only if \(B=B^*\).