Finite matrix topologies

A filter F of positive-primitive formulae may be used to give a right R-module MR the structure MF of a topological abelian group. The topology is called a finite matrix topology if every finite matrix subgroup of MR is closed in MF. It is shown that the pure-injective envelope is functorial on the subcategory of modules for which MF 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 MF is dense in its pure-injective envelope [PE(M)]F. In that case, every R-endomorphism of PE(MR) is determined by its restriction to MR. When M=RR, this gives the pure-injective envelope PE(RR) 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.