Topological ring
A ring that is a topological space, such that the mappings
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are continuous. A topological ring is called separated if it is separated as a topological space (cf. Separation axiom). In this case
is a Hausdorff space. Any subring
of a topological ring
, and also the quotient ring
by an ideal
, is a topological ring. If
is separated and the ideal
is closed, then
is a separated topological ring. The closure
of a subring
in
is also a topological ring. A direct product of topological rings is a topological ring in a natural way.
A homomorphism of topological rings is a ring homomorphism which is also a continuous mapping. If is such a homomorphism, where
is moreover an epimorphism and an open mapping, then
is isomorphic as a topological ring to
. Banach algebras are an example of topological rings. An important type of topological ring is defined by the property that it has a fundamental system of neighbourhoods of zero consisting of some set of ideals. For example, to any ideal
in a commutative ring
one can associate the
-adic topology, in which the sets
for all natural numbers
form a fundamental system of neighbourhoods of zero. This topology is separated if the condition
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is satisfied.
For a topological ring one can define its completion
, which is a complete topological ring, and a separated topological ring
can be imbedded as an everywhere-dense subset in
, which is also separated in this case. The additive group of the ring
coincides with the completion of the additive group of
, as an Abelian topological group.
References
[1] | N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) |
[2] | N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) |
[3] | L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) |
[4] | B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) |
Topological ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topological_ring&oldid=17173