Rings

A ring is a set together with two maps and such that

• is an Abelian groupAbelian group,
• is a monoid,
• is distributive over , meaning for every .

The additive and multiplicative unit elements are denoted and , respectively, and are called identities in the context of rings.

The additive inverse of is denoted .

Let be a ring. Then

for every .

Proof:

1. , and the same for .
2. , and the same for .

Let be a ring. is a subring of if is a ring and . The pair is called a ring extension.

Let be a ring. Then whose elements are called units.