Centred bimodules over prime rings: Closed submodules and applications to ring extensions
Let \(R\) be a prime ring and let \(M\) be a centred \(R\)-bimodule, so that \(M\) has a set of generators which commute with \(R\). A sub-bimodule \(N\) of \(M\) is said to be closed if \(mI \subseteq N\) for \(m \in M\) and \(0 \neq I \triangleleft R\) implies that \(m \in N\). This paper starts by studying these closed sub-bimodules. For example, if \(Q\) is the maximal right ring of quotients of \(R\), then there is a canonical torsion-free extension of \(M\) to a \(Q\)-bimodule \(M^*\), and it is shown that the closed sub- bimodules of \(M\) are in one-to-one correspondence with those of \(M^*\). This result is then applied to study the torsion-free rank of a sub- bimodule of \(M\) and to study non-singular and strongly closed sub- bimodules. Furthermore, these results yield information on the prime ideal structure of centred extensions and of intermediate extensions. Thus, this paper completes and extends previous work of the author [in J. Algebra 148, 1-16 (1992; Zbl 0763.16010)].
scientific article; zbMATH DE number 763203
scientific article