Rings with Polynomial Identities

This post is part of an ongoing series exploring algebraic structures in modern cryptography

Rings are one of the basic mathematical structures used in modern cryptography.

The algebriac structure of a Ring $R$ comes in many shapes and sizes. One of the most basic examples is that of the real numbers.

Ideals

An ideal is a special subset of a ring that absorbs multiplication by any ring element. Formally, for a ring $(R, +, \cdot)$, a subset $I \subseteq R$ is an ideal if:

1. $(I, +)$ is a subgroup of $(R, +)$
2. For all $r \in R$ and $i \in I$, both $r \cdot i \in I$ and $i \cdot r \in I$

Variety

A variety (or algebraic variety) is the set of solutions to a system of polynomial equations. Formally, for polynomials $f_1, f_2, ..., f_n \in k[x_1,...,x_m]$ over a field $k$, the variety $V(f_1,...,f_n)$ is the set of points $(a_1,...,a_m) \in k^m : f_i(a_1,...,a_m) = 0 \text{ for all } i$.