M-injective hulls of topological modules

Abstract

Let [Formula: see text] be a topological ring and [Formula: see text] be a topological [Formula: see text]-module. An [Formula: see text]-module [Formula: see text] is called topological [Formula: see text]-injective if, for every continuous monomorphism [Formula: see text] with [Formula: see text] an open submodule of [Formula: see text], and for every continuous homomorphism [Formula: see text], there exists a continuous homomorphism [Formula: see text] that extends [Formula: see text]. A topological [Formula: see text]-injective hull of [Formula: see text] is a minimal topological [Formula: see text]-injective extension of [Formula: see text]. If [Formula: see text] is a topological module in the category [Formula: see text], then [Formula: see text] has injective hulls in the categories [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], which are not necessarily the same in general. A topological [Formula: see text]-injective hull of [Formula: see text] in [Formula: see text] is the trace of [Formula: see text] in [Formula: see text], where [Formula: see text] is a topological injective hull of [Formula: see text] in [Formula: see text]. If [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] are injective hulls of [Formula: see text] in [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], respectively, then [Formula: see text] and [Formula: see text]. Furthermore, we prove that an infinite direct sum of topological [Formula: see text]-injective hulls is homeomorphic to a topological [Formula: see text]-injective hull of the direct sum of topological [Formula: see text]-modules in [Formula: see text] if the direct sum of topological [Formula: see text]-injective hulls is an open submodule in the direct product of topological [Formula: see text]-injective hulls.