Ring (mathematics)

In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. The branch of abstract algebra which studies rings is called ring theory. For a history and overview of rings see that article.

Contents

1 Formal definition

1.1 Alternative definitions

2 Examples

3 Simple theorems

4 Constructing new rings from given ones

5 See also

Formal definition

A ring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that:

(R, +) is an abelian group with identity element 0:
- (a + b) + c = a + (b + c)
- a + b = b + a
- 0 + a = a + 0 = a
- ∀a ∃(−a) such that a + −a = −a + a = 0
(R, ·) is a monoid with identity element 1:
- 1·a = a·1 = a
- (a·b)·c = a·(b·c)
Multiplication distributes over addition:
- a·(b + c) = (a·b) + (a·c)
- (a + b)·c = (a·c) + (b·c)

As with groups the symbol · is usually omitted and multiplication is just denoted by juxtaposition. Also the standard order of operation rules are used, so that e.g. a+bc is an abbreviation for a+(b·c).

Although ring addition is commutative (i.e. a+b = b+a), note that the commutativity for multiplication (a·b = b·a) is not among the ring axioms listed above. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. Not all rings are commutative.

Also note that an element of a ring need not have a multiplicative inverse. 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 a·b = b·a = 1. If that is the case, then b is uniquely determined by a and we write a⁻¹ = b. The set of all units in R forms a group under ring multiplication; this group is denoted by U(R).

Alternative definitions

There are some alternative definitions of rings of which the reader should be aware:

Some authors add the addition requirement that 0 ≠ 1. This omits only one ring: the so called trivial ring, which has only a single element.

A more significant difference is that some authors omit the requirement that a ring have a multiplicative identity. Those rings which do have multiplicative identities are then called unital rings, unitary rings, or simply 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 article all rings are assumed to be associative and unital unless otherwise stated.

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 and is a subring of the tessarines.
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,...,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.
The Weyl algebra over the field k is generated by 2 elements x and y subject to the relation xy-yx=1.
If G is a group and R is a ring the group ring of G over R is a free module over having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.
The free algebra on a set of indeterminates over the ring R is a further example of a noncommutative ring provided there is more than one indeterminate.
The set of endomorphisms of an object in an abelian category is a ring.
The path algebra of a quiver is another useful noncommutative 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

Constructing new rings from given ones

If a subset S of a ring R is itself a ring with the same operations (restricted to S), and the identity element 1 of R is contained in S, then S is called a subring of R.
The center of a ring R is the set of elements of R that commute with every element of R; that is, c lies in the center if cr=rc for every r in R. The center is a subring of R. We say that a subring S of R is central if it is a subring of the center 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) = (r₁r₂, s₁s₂).

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) = (ab) + 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.