Closure Operations in Module Categories

摘要

Singular left modules over an associative ring A are characterized by the fact that the annihilator of every element is an essential left ideal. These modules play an important part in torsion theory. What can be said about A-bimodules whose elements are annihilated by (two-sided) essential ideals? It is shown that for semiprime rings A this property characterizes bimodules which are singular in the category σ[A] of bimodules subgenerated by A. Based on this observation the closure operations on bimodules studied by M. Ferrero for prime and semiprime rings A are related to the singular torsion theory in σ[A]. In this context we give some characterizations of strongly prime rings. Our methods also apply to non-associative rings.