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)?
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".
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 $\mathbb{R}^{\mathbb N}/\text{MAX}$.
