Definition
A ring is a set $${\displaystyle R}$$ equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. … See more

Prime ideals
As was mentioned above, $${\displaystyle \mathbb {Z} }$$ is a unique factorization domain. … See more

A ring is called local if it has only a single maximal ideal, denoted by m. For any (not necessarily local) ring R, the localization
at a prime ideal p is local. This localization reflects the geometric properties of Spec R "around p". Several notions and problems in commutative … See more

In contrast to fields, where every nonzero element is multiplicatively invertible, the concept of divisibility for rings is richer. An element $${\displaystyle a}$$ of ring $${\displaystyle R}$$ is called a unit if it possesses a multiplicative inverse. Another particular … See more

Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For … See more

A ring homomorphism or, more colloquially, simply a map, is a map f : R → S such that
These conditions … See more

There are several ways to construct new rings out of given ones. The aim of such constructions is often to improve certain properties of the … See more