A module topology is the finest topology such that
scalar multiplication and addition are
continuous. A finitely generated module topology is a
topological ring. Note that this general definition of a module topology does not need to have a ring structure, it merely needs existence of addition and scalar multiplication. A
topological vector space is a topological module over a
topological field. An
abelian topological group can be considered as a topological module over \Z, where \Z is the
ring of integers with the
discrete topology. A topological ring is a topological module over each of its
subrings. A more complicated example is the I-
adic topology on a ring and its modules. Let I be an
ideal of a ring R. The sets of the form x + I^n for all x \in R and all positive integers n, form a
base for a topology on R that makes R into a topological ring. Then for any left R-module M, the sets of the form x + I^n M, for all x \in M and all positive integers n, form a base for a topology on M that makes M into a topological module over the topological ring R. ==See also==