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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 + In, 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 + InM, 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.
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