We can make the followings steps towards the answer of the question for locally compact rings.

We begin from compact Hausdorff rings.

**Theorem 1.** [War, Theorem 1] Let $A$ be a compact  Hausdorff ring, and let $\mathfrak r$ be the radical of $A$. Every ideal of $A$  is  closed if and  only  if $A$  satisfies  the ascending  chain  condition  on  ideals and  every  principal  ideal  of $A$ is closed.  Under these circumstances,  the  topology of $A$  is  the  $\mathfrak r$-adic  topology,  either  $A=\mathfrak r$  or $A/\mathfrak r$  is  isomorphic  to  the  Cartesian product of a  finite family  of finite  simple  rings,  and  consequently  there are  only finitely  many  regular  maximal,  regular  maximal  left,  or  regular  maximal  right ideals in $A$.

Moreover, there is the following structural theorem.

**Theorem 2.** [War, Theorem 2] Let $A$ be a  topological  ring with  identity,  and  let $\mathfrak r$  be  the  radical of $A$. The following  conditions  are equivalent.

1°  $A$  is  compact Hausdorff,  and  every  left  ideal  of $A$  is  closed.

2°  $A$  is  compact Hausdorff and  Noetherian.

3°  $A$  is compact Hausdorff and satisfies the ascending chain condition on closed  left ideals.

4°  $A$  is  Noetherian,  the  topology  of $A$  is  the  $\mathfrak r$-adic  topology,  $A$  is  complete for  that  topology, $\bigcap_{n=1}^\infty \mathfrak r^n=\{0\}$, and $A/\mathfrak r$  is a  finite  ring.

Now we proceed to locally compact rings. In [Kap] a subset $S$ of a topological ring is called *algebraically nilpotent* if for some $n$, $S^n=0$.

**Theorem 3.** [Kap, Theorem 2] A  locally  compact  ring with  no algebraically  nilpotent ideals  is the  direct  sum  of  a connected  ring  and a totally  disconnected  ring. The former  is a semi-simple  algebra  of  finite  order  over  the  real  numbers.

Since Romain Gicquaud's [comment](https://mathoverflow.net/questions/454884/are-topological-pids-noetherian#comment1177754_454894) rules out the connected ring case, it remains to consider locally compact totally disconnected rings. For this case in [Kap, Lemma 4] is provided the following structural lemma.

**Lemma 4.**  A locally  compact  totally  disconnected  ring  *A* has  a system of  neighborhoods  of  *0* which  are compact  open  subrings.

*References*

[Kap] Irving Kaplansky, *[Locally compact rings](https://www.jstor.org/stable/2372343)*, Am. J. Math. **70**:2 (Apr 1948) 447-459.

[War] Seth Warner, *[Compact rings](https://link.springer.com/article/10.1007/BF01452361)*, Mathematische Annalen **145** (Feb 1962) 52-63.