Centred bimodules over prime rings: Closed submodules and applications to ring extensions Original Research Article

Let M be a centred bimodule over a prime ring R. In this paper we define and study a very useful class of sub-bimodules of M: the class of closed sub-bimodules. There is a canonical torsion-free extension of M to a Q-bimodule M* which is always free over Q, where Q is the complete ring of right quotients of R. We prove that closed sub-bimodules of M are in one-to-one correspondence with closed sub-bimodules of M*. The results are applied to study the torsion-free rank of a sub-bimodule of M and to study non-singular and strongly closed sub-bimodules. Also, the results are applied to study prime ideals in centred extensions and intermediate extensions. In particular, we complete and extend the results obtained in [M. Ferrero, J. Algebra 148 (1992), 1–16].