How does ring theory connects with analysis? I'm student who is interested in analysis but now i'm taking some graduate algebra courses.
I have to say that i enjoy a lot algebra especially ring theory.
I would like a lot to find and study aspects in analysis combined with ring theory and advanced algebra.
I know a field in analysis that uses ring and module theory.It's the theory of Banach algrbras.
Are there any more fields in analysis that use ring theory (and generally advanced algebra)?
 A: C*-algebras are important in  functional analysis.
A: 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.
A: 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".
A: One natural connection is with infinitesimal analysis, via the construction of the hyperreals as the quotient by a suitable maximal ideal MAX of the ring of sequences of real numbers, or in formulas $\mathbb{R}^{\mathbb N}/\text{MAX}$.
