Quotients of topological rings

Question

Let $\varphi\colon R\to S$ be a surjective ring homomorphism and let $R$ be a topological ring.

Is there some nice characterization of the finest topology on $S$ for with both $S$ becomes a topological ring and $\varphi$ becomes continuous?

If needs be, further conditions can be imposed on $R$, $S$ and/or $\varphi$.

Thanks for any help!

Any ring $S$ is a topological ring under the trivial topology. Perhaps you are looking for something more "profound"? — hardmath, Aug 04 '14 at 04:14
I rewrote the question. However I doubt the answer to what I have written now is much more profound than the "trivial topology" answer was. — A Rock and a Hard Place, Aug 04 '14 at 04:35
I think this is a more interesting Question. I suggest "finest" topology rather than largest, in the wording. — hardmath, Aug 04 '14 at 04:43
Here's a note about the case (as here) of the quotient of a topological ring. A natural conjecture is that the induced topology by the quotient map is what you describe. — hardmath, Aug 04 '14 at 04:51

score 4 · Accepted Answer · answered Aug 04 '14 at 11:04

I think that hardmath's conjecture in comments that the answer is the quotient topology is correct.

Give $S$ the quotient topology (i.e., a subset $U\subseteq S$ is open if and only if $\varphi^{-1}(U)\subseteq R$ is open), and let $I=\ker(\varphi)$.

The quotient topology is the finest topology on $S$ for which $\varphi$ is continuous, so if we can show that it makes $S$ into a topological ring (i.e., all the ring operations are continuous) then we're done.

The key observation is that $\varphi$ is an open map: if $V\subseteq R$ is open, then $$\varphi^{-1}\left(\varphi\left(V\right)\right)=\bigcup_{x\in I}(V+x)$$ is a union of translates of $V$, and therefore open, and so $\varphi(V)\subseteq S$ is open.

It now easily follows that $$\varphi\times\varphi:R\times R\to S\times S$$ is also an open map, since $(\varphi\times\varphi)(V_1\times V_2)=\varphi(V_1)\times\varphi(V_2)$ is open in $S\times S$ if $V_1$ and $V_2$ are open in $R$.

Consider the multiplication maps $\mu:R\times R\to R$ and $\overline{\mu}:S\times S\to S$, and let $U\subseteq S$ be open. Then $$\overline{\mu}^{-1}(U)=\left(\varphi\times\varphi\right)\left(\mu^{-1}\varphi^{-1}(U)\right)$$ is open. So $\overline{\mu}$ is continuous, and similar proofs show that the other ring operations on $S$ are continuous.

Quotients of topological rings

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