Math and science::Algebra::Aluffi
Ring. Definition.
Ring
An abelian group
- is associative
- has a two-sided identity
- distributes with respect to the preexisting group operation
The first two requirements for
Monoid
A set with an associative operation is a semigroup; a semigroup with an identity is a monoid; and a group is a monoid where all elements have two-sided inverses.Distributive law
The distributive law looks quite similar in form to the requirements of a group homomorphism. This is emergent from how rings can be thought of as the group of endomorphisms of an abelian group, with the added operation of function composition.
Example
Important examples of rings include:
- The zero ring
. - The integers with multiplication,
. - Cyclic groups with added multiplication,
. - The reals with multiplication,
. - Square matrices of any size, with entries from a ring.
- Polynomial rings.
- Monoid rings (generalization of ploynomial rings).
Some of the rings above are commutitive in their second operation, some are not. Some have inverses with respect to the second operation, some do not.