A ring is a set together with two binary operations, denoted and , such that
and for all (associative law)
for all (commutative law)
There exists an element such that for all (additive identity)
For all , there exists such that (additive inverse)
and for all (distributive law)
Equivalently, a ring is an abelian group 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 where is the additive inverse of .
We say has a multiplicative identity if there exists an element such that for all . Alternatively, one may say that 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 does have a multiplicative identity, then a multiplicative inverse of an element is an element such that . An element of that has a multiplicative inverse is called a unit of .
A ring is commutative if for all .
|Date of creation||2013-03-22 11:48:40|
|Last modified on||2013-03-22 11:48:40|
|Last modified by||djao (24)|
|Defines||ring with unity|