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 π0(R) = R/( ⋂︁ n⩾1 In) = t(R) where π0(R) is the space of connected components of R and t(R) 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
