ring
A ring is a set $R$ together with two binary operations, denoted $+:R\times R\u27f6R$ and $\cdot :R\times R\u27f6R$, such that

1.
$(a+b)+c=a+(b+c)$ and $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ for all $a,b,c\in R$ (associative law)

2.
$a+b=b+a$ for all $a,b\in R$ (commutative law)

3.
There exists an element $0\in R$ such that $a+0=a$ for all $a\in R$ (additive identity)

4.
For all $a\in R$, there exists $b\in R$ such that $a+b=0$ (additive inverse)

5.
$a\cdot (b+c)=(a\cdot b)+(a\cdot c)$ and $(a+b)\cdot c=(a\cdot c)+(b\cdot c)$ for all $a,b,c\in R$ (distributive law)
Equivalently, a ring is an abelian group^{} $(R,+)$ together with a second binary operation $\cdot $ such that $\cdot $ is associative and distributes over $+$. Additive inverses are unique, and one can define subtraction in any ring using the formula $ab:=a+(b)$ where $b$ is the additive inverse of $b$.
We say $R$ has a multiplicative identity^{} if there exists an element $1\in R$ such that $a\cdot 1=1\cdot a=a$ for all $a\in R$. 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 $a\in R$ is an element $b\in R$ such that $a\cdot b=b\cdot a=1$. An element of $R$ that has a multiplicative inverse is called a unit of $R$.
A ring $R$ is commutative if $a\cdot b=b\cdot a$ for all $a,b\in R$.
Title  ring 
Canonical name  Ring 
Date of creation  20130322 11:48:40 
Last modified on  20130322 11:48:40 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  19 
Author  djao (24) 
Entry type  Definition 
Classification  msc 1600 
Classification  msc 2000 
Classification  msc 1300 
Classification  msc 81P10 
Classification  msc 81P05 
Classification  msc 81P99 
Related topic  ExampleOfRings 
Related topic  Subring 
Related topic  Semiring^{} 
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 