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ring (Definition)

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. $ (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$.



"ring" is owned by djao. [ full author list (2) ]
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See Also: examples of rings, subring, semiring, group, associates

Also defines:  multiplicative identity, multiplicative inverse, ring with unity, unit, ring addition, ring multiplication, ring sum, ring product, unital ring, unitary ring

Attachments:
uniqueness of additive inverse in a ring (Theorem) by alozano
zero times an element is zero in a ring (Theorem) by alozano
minus one times an element is the additive inverse in a ring (Theorem) by alozano
uniqueness of additive identity in a ring (Theorem) by alozano
examples of rings (Example) by matte
Klein 4-ring (Definition) by pahio
left and right unity of ring (Definition) by rspuzio
additive inverse of one element times another element is the additive inverse of their product (Theorem) by cvalente
additive inverse of a sum in a ring (Theorem) by rspuzio
additive inverse of an inverse element (Result) by Mathprof
ring hierarchy (Topic) by Algeboy
inverses in rings (Topic) by Wkbj79
hollow matrix rings (Example) by Algeboy
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Cross-references: subtraction, distributes over, abelian group, distributive law, inverse, identity, additive, commutative law, associative, binary operations
There are 653 references to this entry.

This is version 14 of ring, born on 2001-10-19, modified 2006-11-22.
Object id is 354, canonical name is Ring.
Accessed 50146 times total.

Classification:
AMS MSC: 13-00 (Commutative rings and algebras :: General reference works )
 16-00 (Associative rings and algebras :: General reference works )
 20-00 (Group theory and generalizations :: General reference works )
Pending Errata and Addenda
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Discussion
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Ring definition ( by djao ) by jimcp on 2007-06-25 15:23:54
In Dummit & Foote it is proved that the following definition of a ring "A ring (R,+, ·) is a group (R, +) with an additional associative multiplication · with identity, which distributes over the addition." implies commutativity in the additive group. Yet authors persist to include commutativity as an axiom in Ring Theory.

Any comments?
[ reply | up ]
group? by Gunnar on 2003-11-13 18:07:49
Can't a ring (R,+,*) be described as a commutative group (R,+) and a * (closed) operation?
[ reply | up ]
commutative by antizeus on 2001-10-19 00:34:49
if it were me I'd get rid of the sentence about commutative rings. I'd probably also express it in terms of additive abelian groups and multiplicative semigroups or monoids, so maybe you shouldn't listen to me.
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