Discrete and Profinite Modules

Abstract

A profinite ring Λ is an inverse limit of an inverse system {Λ _i ,φ _ij } of finite rings. We always assume that rings have an identity element, denoted usually by 1, and that homomorphisms of rings send identity elements to identity elements. A profinite ring Λ is plainly a compact, Hausdorff and totally disconnected topological ring; the converse is also true, as we indicate in Proposition 5.1.2 below. It is clear that a profinite ring admits a fundamental system of neighborhoods of 0 consisting of open (two-sided) ideals (this follows from a result analogous to Lemma 2.1.1).