# What are some of the most classical rings studied in Algebraic Geometry?

I am curious what are the typical rings studied in Algebraic Geometry?

I am aware that the polynomial ring $K[x_1,\dots,x_n]$, where $K$ is a field, is of great interest.

Are there any other classical rings that are studied?

Thanks.

Update: I am also interested in PIDs studied in Algebraic Geometry. Are there any classical ones?

• Local rings are hugely important. In particular, localisations of quotients of $K[x_1, \dots, x_n]$. – Mr. Chip Nov 16 '17 at 15:45
• I guess for a finite matrix group $G\subseteq GL_n(k)$ the ring of invariants $k[x_1,...,x_n]^G$ is also of great interest. – Levent Nov 16 '17 at 23:27

Surely the field of rational functions $K(X)$ for a field $K$, power-series rings, and Laurent series rings also qualify.
Based on my knowledge of the history of algebraic geometry (what little of it I have, I learned from Dieudonne's paper/lecture on the topic$^\ast$), at least the rational functions have been important in algebraic geometry since the 19th century.
$^\ast$ Dieudonné, Jean. "The historical development of algebraic geometry." The American Mathematical Monthly 79.8 (1972): 827-866.