Finite matrix topologies

Abstract

A filter F of positive-primitive formulae may be used to give a right R-module M R the structure M F of a topological abelian group. The topology is called a finite matrix topology if every finite matrix subgroup of M R is closed in M F . It is shown that the pure-injective envelope is functorial on the subcategory of modules for which M F is dense in its pure-injective envelope. We call a right R-module almost pure-injective if there is a filter F with respect to which the topological abelian group M F is dense in its pure-injective envelope [ PE ( M ) ] F . In that case, every R-endomorphism of PE ( M R ) is determined by its restriction to M R . When M = R R , this gives the pure-injective envelope PE ( R R ) a ring structure extending that of R, and the proof of this result suggests that this ring is the pure variation of the ring of quotients of a nonsingular ring.