Online Encyclopedia
Ring (mathematics)
In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers.
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History
See Ring theory
Definition and notation
A ring is an abelian group (R, +), together with a second binary operation * such that for all a, b and c in R,
- a * (b*c) = (a*b) * c [Associativity]
- a * (b+c) = (a*b) + (a*c) [* Pre-Distributive over +]
- (a+b) * c = (a*c) + (b*c) [* Post-Distributive over +]
and such that there exists a multiplicative identity, or unity, that is, an element 1 so that for all a in R,
- a*1 = 1*a = a
(Some authors omit the requirement for a multiplicative identity, and call those rings which do have multiplicative identities unitary rings, unital rings or rings with a 1. 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.)
Note that the commutative law,
- a*b = b*a for all a,b ∈ R
is not among the ring axioms listed above; rings that satisfy this law (such as the ring of integers) are called commutative rings. Not all rings are commutative; see, for example, Matrix rings, described below.
The identity element with respect to + is called the zero element of the ring and written as 0. The symbol * is usually omitted from the notation, and the standard order of operation rules are used, so that e.g. a+bc is an abbreviation for a+(b*c). The additive inverse of the element x in a ring is written as -x.
In a ring we have 0=1 if and only if we are dealing with the trivial ring {0} with a single element.
An element a in a ring is called a unit if it is invertible with respect to multiplication, i.e., if there is an element b in the ring such that
- ab = ba = 1
If that is the case, then b is uniquely determined by a and we write a-1 = b.
Examples
- The motivating example is the ring of integers with the two operations of addition and multiplication. This is a commutative ring.
- The rational, real and complex numbers form rings, in fact they are even fields. These are likewise commutative rings.
- More generally, every field is a commutative ring.
- If n is a positive integer, then the set Z/nZ of integers modulo n forms a ring with n elements (see modular arithmetic).
- The split-complex plane D is a ring useful in modern physics.
- 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.
- For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations. For n=1, this matrix ring is just (isomorphic to) R itself. For n>2, this matrix ring is an example of a noncommutative ring (unless R is the trivial ring).
- The trivial ring {0} has only one element which serves both as additive and multiplicative identity.
- 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.
- The set of formal power series R[[X1,...,Xn]] over a commutative ring R is a ring.
- The set of all functions in n complex variables holomorphic at the origin is a ring.
Simple theorems
From the axioms, one can immediately deduce that, for all elements a and b of a ring, we have
- 0a = a0 = 0
- (-1)a = -a
- (-a)b = a(-b) = -(ab)
- (ab)-1=b-1 a-1 if both a and b are invertible, and hence the set of all invertible elements in a ring is closed under multiplication * and forms a group, the group of units of the ring.
Constructing new rings from given ones
- If a subset S of a ring (R,+,*) together with the operations + and * restricted on S is itself a ring, and the identity element 1 of R is contained in S, then S is called a subring of (R,+,*).
- The centre of a ring R is the set of elements of R that commute with every element of R; that is, c lies in the centre if cr=rc for every r in R. The centre is a subring of R. We say that a subring S of R is central if it is a subring of the centre of R.
- The direct sum of two rings R and S is the cartesian product R×S together with the operations
- (r1, s1) + (r2, s2) = (r1+r2, s1+s2) and
- (r1, s1) * (r2, s2) = (r1*r2, s1*s2).
- Given a ring R and an ideal I of R, the quotient ring (or factor ring) R/I is the set of cosets of I together with the operations
- (a+I) + (b+I) = (a+b) + I and
- (a+I) * (b+I) = (a*b) + I.
- Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring provided S is a central subring of R and T.
Glossary and related topics
See Glossary of ring theory for more definitions in ring theory.