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Ring (mathematics)

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.

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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

  • The motivating example is the ring of integers with the two operations of addition and multiplication.
  • The rational, real and complex numbers form rings, in fact they are even fields.
  • If n is a positive integer, then the set Zn of integers modulo n forms a ring with n elements (see modular arithmetic).
  • The set of all continuous real-valued functions defined on the interval [a, b] forms a ring (even an associative algebra). The operations are addition and multiplication of functions.
  • The set of all polynomials over some common coefficient ring forms a ring.
  • The set of all square n-by-n matrices, where n is fixed, forms a ring with matrix addition and matrix multiplication as operations.
  • If G is an abelian group, then the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms.
  • If S is a set, then the power set of S becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.

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

  • If a subset H of a ring (R,+,*) together with the operations + and * restricted on H is itself a ring, then it is called a subring of (R,+,*).
  • The direct sum of two rings (G,+,*) and (H",#,•) is the set G×H together with the operations
(g1,h1)^(g2,h2) = (g1+g2,h1#h2) and
(g1,h1)(g2,h2) = (g1*g2,h1h2).
  • Given a ring R and and ideal I, the factor ring is the set of cosets of R/I together with operations gI+hI=(g+h)I and (gI)(hI)=ghI.

Glossary and related topics

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



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