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Ring (mathematics)

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In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. The branch of mathematics which study rings is called ring theory.

Table of contents

Definition

A ring is an abelian group (R, +), together with a second binary operation * such that for all a, b and c in R,
<math>a * (b*c) = (a*b) * c</math>
<math>a * (b+c) = (a*b) + (a*c)</math>
<math>(a+b) * c = (a*c) + (b*c)</math>

and such that there exists a multiplicative identity, or unity, that is, an element 1 so that for all a in R,

<math>a*1 = 1*a = a</math>

Many authors omit the requirement for a multiplicative identity, and call those rings which do have multiplicative identities unitary rings. Similarly, the requirement for the ring multiplication to be associative is sometimes dropped, and rings in which the associative law holds are called associative rings. In this encyclopedia, associativity and the existence of a multiplicative identity are taken to be part of the definition of a ring.

The symbol * is usually omitted from the notation, so that a * b is just written a'b.

Examples

First consequences

From the axioms, one can immediately deduce that

<math>0a = a0 = 0</math>
<math>(-1)a = -a</math>
<math>(-a)b = a(-b) = -(ab)</math>

for all elements a and b in R. Here, 0 is the neutral element with respect to addition +, and -x stands for the additive inverse of the element x in R.

An element a in a ring is called a unit if it is invertible, i.e., there is an element b such that

<math>ab = ba = 1</math>
If that is the case, then b is uniquely determined by a and we write a-1 = b. The set of all invertible elements in a ring is closed under multiplication * and therefore forms a group, the group of units of the ring. If both a and b are invertible, then we have
<math>(ab)^{-1}=b^{-1}a^{-1}</math>

Constructing new rings from given ones

(g1,h1)^(g2,h2) = (g1+g2,h1#h2) and
(g1,h1)(g2,h2) = (g1*g2,h1h2).

Glossary and related topics

See Glossary of ring theory for more definitions in ring theory.

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