Why do we need topological rings?

Question

Today I had an interview and a professor asked me my research topic, which is topological rings. The follow up question was that topological rings are a richer structure so it must solve some problems that either rings or topological spaces alone don't solve. I was blank on this question as I never wondered on this question so I am asking here now that what are some properties that doesn't hold for algebraic rings but are enjoyed in topological rings. Also the same for topological spaces.

Answer 1

It's a bit silly to draw a line between "topological rings" and "algebraic rings" as if rings could somehow be anti-topological. Every ring can be made into a topological ring with the discrete topology.

The real question is: what is gained if a particular ring has a useful topology on it?

There is one oft-noted quality that topological rings have that infinite sums and infinite products have a chance to be defined (if one has a notion of convergence because of the topology, then there is hope that the infinite sum/product is the limit of partial sum/products.)

And another thing is that perhaps there are properties that are valid for subsets which will be preserved for the closure of the subset (the closure not being defined without a topology.) For example, is a function defined by a real power series (within its domain of convergence) continuous? Sure, if you work with the natural product topology wherein polynomials are absolutely continuous. Each power series is a limit of polynomials, and the limit of absolutely continuous functions is continuous.

Why not just go find something Kaplansky wrote and read it? That should convince you of the usefulness of topological algebra in general.

Answer 2

As noted in rschwieb's answer any ring can be made into a topological ring using the discrete topology, so there aren't any properties expressible in the language of ring theory that hold in topological rings but don't hold in rings in general.

However, topological rings enjoy topological properties that don't hold in arbitrary topological spaces. The most obvious one is homogeneity: in a topological ring $R$, given any two elements $x, y$ of $R$ there is a self-homeomorphism of $R$ that maps $x$ to $y$.