Ring

A ring $(R, +, \cdot, 0, 1)$ is a set equipped with two binary operations — addition and multiplication — such that $(R, +, 0)$ is an abelian group, $(R, \cdot, 1)$ is a monoid, and multiplication distributes over addition on both sides.

The integers $\mathbb{Z}$, polynomial rings, and matrix rings are standard examples. A ring is commutative if $a \cdot b = b \cdot a$ for all elements. The study of commutative rings and their ideals forms the foundation of algebraic geometry and algebraic number theory.

Rings connect to lattice theory through the notion of residuation: the Heyting implication in a Heyting algebra satisfies an adjunction analogous to the division-like operation in a ring.