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Math and science::Algebra::Aluffi

Ring. Definition.

Ring

An abelian group \( (R, +) \) when endowed with a second binary operation \( \cdot \) is said to form a ring \( (R, +, \cdot) \) if the second operation:

  1. is associative
  2. has a two-sided identity
  3. distributes with respect to the preexisting group operation

The first two requirements for \( \cdot \) make \( (R, \cdot) \) a monoid (and not a group).


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, \( (\mathbb{Z}, +, \cdot) \).
  • Cyclic groups with added multiplication, \( (\mathbb{Z}/n\mathbb{Z}, +, \cdot) \).
  • The reals with multiplication, \( (\mathbb{R}, +, \cdot) \).
  • 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.