In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a topological module is a module over a topological ring

Topological ring

In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology.- General comments :...

 such that scalar multiplication

Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra . In an intuitive geometrical context, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector without changing its direction...

 and addition are continuous.

Examples

A topological vector space

Topological vector space

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

 is a topological module over a topological field.

An abelian

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

 topological group

Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...

 can be considered as a topological module over Z, where Z is the ring of integers

Ring of integers

In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...

 with the discrete topology.

A topological ring is a topological module over each of its subring

Subring

In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...

s.

A more complicated example is the I-adic topology on a ring and its modules. Let I be an ideal

Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

 of a ring R. The sets of the form , for all x in R and all positive integers n, form a base

Base (topology)

In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...

for a topology on R that makes R into a topological ring. Then for any left R-module M, the sets of the form , 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.