ring
A ring is a set together with two binary operations, denoted and , such that
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1.
and for all (associative law)
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2.
for all (commutative law)
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3.
There exists an element such that for all (additive identity)
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4.
For all , there exists such that (additive inverse)
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5.
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 .
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 |