Definition
A ring is a set $${\displaystyle R}$$ equipped with two binary operations, i.e. operations … See more

Prime ideals
As was mentioned above, $${\displaystyle \mathbb {Z} }$$ is a unique factorization domain. … 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 ring so as to make it more readily … 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 … See more

Many of the following notions also exist for not necessarily commutative rings, but the definitions and properties are usually more complicated. For example, all ideals in a commutative ring are automatically two-sided, which simplifies the situation considerably. 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