A ring is a set R together with two binary operations, denoted +:R×RR and :R×RR, such that

  1. 1.

    (a+b)+c=a+(b+c) and (ab)c=a(bc) for all a,b,cR (associative law)

  2. 2.

    a+b=b+a for all a,bR (commutative law)

  3. 3.

    There exists an element 0R such that a+0=a for all aR (additive identity)

  4. 4.

    For all aR, there exists bR such that a+b=0 (additive inverse)

  5. 5.

    a(b+c)=(ab)+(ac) and (a+b)c=(ac)+(bc) for all a,b,cR (distributive law)

Equivalently, a ring is an abelian groupMathworldPlanetmath (R,+) together with a second binary operation such that is associative and distributes over +. Additive inverses are unique, and one can define subtraction in any ring using the formula a-b:=a+(-b) where -b is the additive inverse of b.

We say R has a multiplicative identityPlanetmathPlanetmath if there exists an element 1R such that a1=1a=a for all aR. Alternatively, one may say that R is a ring with unity, a unital ring, or a unitary ring. Oftentimes an author will adopt the convention that all rings have a multiplicative identity. If R does have a multiplicative identity, then a multiplicative inverse of an element aR is an element bR such that ab=ba=1. An element of R that has a multiplicative inverse is called a unit of R.

A ring R is commutative if ab=ba for all a,bR.

Title ring
Canonical name Ring
Date of creation 2013-03-22 11:48:40
Last modified on 2013-03-22 11:48:40
Owner djao (24)
Last modified by djao (24)
Numerical id 19
Author djao (24)
Entry type Definition
Classification msc 16-00
Classification msc 20-00
Classification msc 13-00
Classification msc 81P10
Classification msc 81P05
Classification msc 81P99
Related topic ExampleOfRings
Related topic Subring
Related topic SemiringMathworldPlanetmath
Related topic Group
Related topic Associates
Defines multiplicative identity
Defines multiplicative inverse
Defines ring with unity
Defines unit
Defines ring addition
Defines ring multiplication
Defines ring sum
Defines ring product
Defines unital ring
Defines unitary ring