Some notes on topological rings and their groups of units
Abolfazl Tarizadeh
TL;DR
Addresses the problem of when the unit group $R^{\ast}$ of a topological ring forms a topological group, introducing the notion of absolute topological rings. Shows that any commutative ring with an $I$-adic topology is absolute and analyzes the topology $\mathscr{T}_{f}$ on $R^{\ast}$ given by $a\mapsto(a,a^{-1})$, establishing that $R^{\ast}$ is a topological group iff $\mathscr{T}=\mathscr{T}_{f}$. Demonstrates that monomial and polynomial maps are continuous in these topological ring settings and corrects prior results in the literature (Koh kwangil, Ursul) in the $I$-adic framework; proves an identification $\pi_{0}(R)=R/(\bigcap_{n\ge1}I^{n})=t(R)$ for the $I$-adic topology. Also discusses a gap in the standard proof that every epimorphism in the category of Hausdorff spaces has dense image and provides a corrected argument.
Abstract
If $R$ is a topological ring then $R^{\ast}$, the group of units of $R$, with the subspace topology is not necessarily a topological group. This leads us to the following natural definition: By an \emph{absolute topological ring} we mean a topological ring such that its group of units with the subspace topology is a topological group. We prove that every commutative ring with the $I$-adic topology is an absolute topological ring. Next, we prove that if $I$ is an ideal of a ring $R$ then for the $I$-adic topology over $R$ we have $π_{0}(R)=R/(\bigcap\limits_{n\geqslant1}I^{n})=t(R)$ where $π_{0}(R)$ is the space of connected components of $R$ and $t(R)$ is the space of irreducible closed subsets of $R$. We observed that the main result of Koh \cite{kwangil} as well as its corrected form \cite[Chap II, \S12, Theorem 12.1]{Ursul} are not true, and then we corrected this result in the right way. In the Wikipedia pages, it is claimed that ``the identity component of a topological group is always a characteristic subgroup'', we also provide a counterexample to this claim. Finally, we fix a gap in the proof of the fact that every epimorphism of the category of Hausdorff topological spaces has a dense image.