Ring Theory
A ring is a set equipped with two operations (usually referred to as addition and multiplication) that satisfy certain properties: there are additive and multiplicative identities and additive inverses, addition is commutative, and the operations are associative and distributive.
The study of rings has its roots in algebraic number theory, via rings that are generalizations and extensions of the integers, as well as algebraic geometry, via rings of polynomials. These kinds of rings can be used to solve a variety of problems in number theory and algebra; one of the earliest such applications was the use of the Gaussian integers by Fermat, to prove his famous two-square theorem. There are many examples of rings in other areas of mathematics as well, including topology and mathematical analysis.
Definition and Classification
A ring is a set R together with two operations (+) and (⋅) satisfying the following properties (ring axioms):
(1) R is an abelian group under addition. That is, R is closed under addition, there is an additive identity (called 0), every element a ∈ R has an additive inverse −a ∈ R, and addition is associative and commutative.
(3) Multiplication distributes over addition:
A ring is usually denoted by (R,+,⋅) and often it is written only as RR when the operations are understood.
Notes:
(1) There are two further requirements one might impose on a ring SS that lead to interesting classes of rings. For instance, if multiplication is commutative, the ring is called a commutative ring. The theory of commutative rings differs quite significantly from the the theory of non-commutative rings; commutative rings are better understood and have been more extensively studied. Most of the examples and results in this wiki will be for commutative rings. Again there may be an element 1 in R such that for all elements a in R, a.1=1.a=a. If such an element exists, we call it the unity of the ring, and the ring is called a ring with unity. Else it is called a ring without unity or a "rng" (a ring without i).
(2) If R is a commutative ring and a,b,c∈R such that a,b ≠ 0 and a.b=c ,then a and b are said to be divisors of c. If in a commutative ring R with unity, there is no divisor of the additive identity, i.e. 0, then R is said to be an integral domain. Thus a commutative ring R with unity is said to be an integral domain if for all elements a,b in R, a.b=0 implies either a=0 or b=0.
(3) If every nonzero element in a commutative ring with unity has a multiplicative inverse as well, the ring is called a field. Fields are fundamental objects in number theory, algebraic geometry, and many other areas of mathematics. If every nonzero element in a ring with unity has a multiplicative inverse, the ring is called a division ring or a skew field. A field is thus a commutative skew field. Non-commutative ones are called strictly skew fields.
Examples of Ring
This section lists many of the common rings and classes of rings that arise in various mathematical contexts.
(1) The ring Z of integers is the canonical example of a ring. It is an easy exercise to see that Z is an integral domain but not a field.
(2) There are many other similar rings studied in algebraic number theory, of the form Z[α], where α is an algebraic integer. For example, Z[√2] ={ a+b√2 : a,b∈Z} is a ring, an integral domain, to be precise. Also we have the ring of Gaussian integers Z[i] ={ a+bi : a,b∈Z}, where i is the imaginary unit.
(3) If R is a ring, then so is the ring R[x] of polynomials with coefficients in R. In particular, when R= Z/pZ is the finite field with p elements, R[x] has many similarities with Z.
For example, there is a Euclidean algorithm and hence unique factorization into irreducibles. See the introduction to algebraic number theory for details.
More generally, if X is a set and R is a ring, the set of functions from X to R is a ring, with the natural operations of pointwise addition and multiplication of functions. For many sets X, this ring has many interesting subrings constructed by restricting to functions with properties that are preserved under addition and multiplication. If X = R = R , for instance, there are subrings of continuous functions, differentiable functions, polynomial functions, and so on.