Some notes on topological rings and their groups of units

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 ${\pi }_0(R)=R/(\bigcap _{n⩾1}{I}^n)=t(R)$ where ${\pi }_0(R)$ 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.

12 pages