ring

A ring is a set $R$ together with two binary operations, denoted $+:R\times R⟶R$ and $\cdot :R\times R⟶R$, 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 $a-b:=a+(-b)$ 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	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	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