Some notes on topological rings and their groups of units

Abstract

If R is a topological ring then $R^{⁎}$, 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 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 (where I is an ideal of the ring).

Next, we prove that if I is an ideal of a ring R then for the I-adic topology over R we have ${\pi }_0\left(R\right)=R/\left(\bigcap _{n⩾1}{I}^n\right)=t\left(R\right)$ where ${\pi }_0\left(R\right)$ is the space of connected components of R and $t\left(R\right)$ is the space of irreducible closed subsets of R.

We also show with an example that the identity component of a topological group is not necessarily a characteristic subgroup.

Finally, we observed that the main result of Koh [3] as well as its corrected form [5, Chap II, §12, Theorem 12.1] is not true, and then we corrected this result in the right way.