Topological ring

2020 Mathematics Subject Classification: Primary: 13Jxx [MSN][ZBL]

A topological ring is a ring $R$ that is a topological space, and such that the mappings \[ (x,y) \mapsto x-y \] and \[ (x,y) \mapsto xy \] are continuous. A topological ring $R$ is called separated if it is separated as a topological space (cf. Separation axiom). In this case $R$ is a Hausdorff space. Any subring $M$ of a topological ring $R$, and also the quotient ring $R/J$ by an ideal $J$, is a topological ring. If $R$ is separated and the ideal $J$ is closed, then $R/J$ is a separated topological ring. The closure $\bar{M}$ of a subring $M$ in $R$ 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 $f:R_1 \rightarrow R_2$ is such a homomorphism, where $f$ is moreover an epimorphism and an open mapping, then $R_2$ is isomorphic as a topological ring to $R_1/\mathrm{Ker}f$. 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 $\mathfrak{m}$ in a commutative ring $R$ one can associate the $\mathfrak{m}$-adic topology, in which the sets $\mathfrak{m}^n$ for all natural numbers $n$ form a fundamental system of neighbourhoods of zero. This topology is separated if the condition \[ \bigcap_n \,\mathfrak{m}^n = 0 \] is satisfied.

For a topological ring $R$ one can define its completion $\hat{R}$, which is a complete topological ring, and a separated topological ring $R$ can be imbedded as an everywhere-dense subset in $\hat{R}$, which is also separated in this case. The additive group of the ring $\hat{R}$ coincides with the completion of the additive group of $R$, as an Abelian topological group.

References

[Bo]	N. Bourbaki, "Elements of mathematics. General topology", Addison-Wesley (1966) (Translated from French)
[Bo2]	N. Bourbaki, "Elements of mathematics. Commutative algebra", Addison-Wesley (1972) (Translated from French)
[Po]	L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian)
[Wa]	B.L. van der Waerden, "Algebra", 1–2, Springer (1967–1971) (Translated from German)