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

Abstract

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].