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.
Table of contents |
2 Definition and notation 3 Examples 4 Simple theorems 5 Constructing new rings from given ones 6 Glossary and related topics |
History
See Ring theoryDefinition 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 +]
- a*1 = 1*a = a
Note that the commutative law,
- a*b = b*a for all a,b ∈ R
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
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 fieldss. 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 set of all continuous real-valued functionss 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[[X_{1},...,X_{n}]] 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
- 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
- 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.