# ring

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

1. 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. 2.

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

3. 3.

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

4. 4.

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

5. 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