How does ring theory connects with analysis?

In mathematical control theory you encounter the ring $\mathbb K^{n\times n}[x]$ of matrix polynomials. Here you can ask questions like is a given matrix polynomial A(x) invertible? Answer: it is if $\det(A(x))\in\mathbb K\setminus\{0\}$; in this case $A(x)$ is called $unimodular$.

Another question that naturally arises is what the simplest representative of $A$ under a the similarity transform $A\to SAT$ with $S$ and $T$ unimodular. This leads to the Smith-Normal-Form.

In complex analysis, one is quite often in touch with ring theory. Consider for example the set of all holomorphic(=analytic) functions on a domain (=non-empty, open, connected subset of $\mathbb{C}$) $G$, i.e. $$ H(G) := \{ f: G \rightarrow \mathbb{C} \, | \, f \text{ holomorphic in } G \} $$

This set, equipped with pointwise addition and multiplication, forms a commutative ring with identity element. In fact, $H(G)$ is even an integral domain. When considering the larger set $M(G)$ of meromorphic functions on $G$, this set is even a field. By using the so-called Weierstrass factorization theorem (see https://en.wikipedia.org/wiki/Weierstrass_factorization_theorem), one can show that in fact $M(G)$ is the quotient field of $H(G)$. See for example Conway, "Functions of One Complex Variable I + II" and the books on complex analysis by Remmert. A nice paper on this topic is for example Royden, "Rings of analytic and meromorphic functions".

C*-algebras are important in functional analysis.