Is every open ideal in topological rings closed ideal?

Question

If we have $I$ as an ideal in topological ring $R$, then $I$ is an ideal in the algebraic sense with the subspace topology. So, $I$ is a subgroup of $R$ as a topological group. From basics topological groups properties, we have every open subgroup of a topological group is closed. Is it right if we say that an open ideal $I$ of topological ring $R$ is a closed ideal? Because open or closed subset depends on those are elements in topology or not. Am I right? When I read books on topological rings, they always use a closed ideal than an open ideal, is there something that I missed? Thank you so much for any help.

Answer 1

Yes that seems to be correct to me. I'm not that familiar with topological rings (more with groups) but if $I$ is an open ideal, in particular it's an open subgroup, and as $R$ is a disjoint union of translates of $I$, like in groups, $I$ is also closed. And still an ideal of course. (and we have that $R$ is topologically disconnected (if $I \neq R$). So looking at closed ideals seems nicer and more general. $R{/}I$ then might be a topological ring again too, I'm not quite sure (but at least it will be $T_1$ again).