A ring is a mathematical concept that is used to describe sets that have two binary operations, addition and multiplication. This notion was first introduced in the mid-19th century by Richard Dedekind and later formalized by Abraham Fraenkel in 1914.
In order to be considered a ring, a set must have certain properties. These properties include:
- a + b = b + a for all elements a and b in the set. This property is known as commutativity of addition.
- (a + b) + c = a + (b + c) for all elements a, b, and c in the set. This property is known as associativity of addition.
- There is an additive identity, 0, such that a + 0 = a for all elements a in the set. This property is known as the existence of an additive identity.
- There is an element -a such that a + (-a) = 0. This property is known as the existence of additive inverses.
- a(bc) = (ab)c and (b+c)a = ba + ca for all elements a, b, and c in the set. These properties are known as associativity and distributivity of multiplication over addition.
From these properties, we can see that a ring is an Abelian group under addition, meaning that the group is commutative. Additionally, the multiplication operation must be associative and left and right distributive over addition. It is important to note that multiplication need not be commutative in a ring. When it is, the ring is called commutative ring.
When a ring does have an identity under multiplication, it is called a unity or an identity. In a commutative ring with unity, not all nonzero elements have a multiplicative inverse. When an element does have a multiplicative inverse, it is called a unit of the ring.
Examples of rings include:
- The set of integers Z under ordinary addition and multiplication is a commutative ring with unity 1. The units of Z are 1 and -1.
- The set of integers modulo n, Zn = {0, 1, . . . , n − 1} under addition and multiplication modulo n is a commutative ring with unity 1. The set of units is U (n ).
- The set of all polynomials in the variable x with integer coefficients under ordinary addition and multiplication is a commutative ring with unity f (x ) = 1.
- The set of even integers under ordinary addition and multiplication is a commutative ring without unity.
- The set of all continuous real-valued functions of a real variable whose graphs pass through the point (1, 0) is a commutative ring without unity under the operations of pointwise addition and multiplication.
It is worth noting that there are also non-commutative examples of rings, such as the set of all square matrices of a fixed size with real entries. The set of all square matrices of a fixed size with complex entries also forms a non-commutative ring.
In addition to the two obvious properties of commutativity and existence of a unity, there is one other essential feature of the integers that rings in general do not enjoy—the cancellation property. To this end, we introduce the concept of integral domains. Integral domains, a particular class of rings that have all three of these properties. Integral domains play a prominent role in number theory and algebraic geometry.
A zero-divisor is a nonzero element a of a commutative ring R such that there is a nonzero element b ∈ R with ab = 0. An integral domain is a commutative ring with unity and no zero-divisors. This means that in an integral domain, a product is 0 only when one of the factors is 0; that is, ab = 0 only when a = 0 or b = 0.
Examples of integral domains include the ring of integers, the ring of Gaussian integers Z [i], the ring of polynomials with integer coefficients Z [x], the ring Z [ √2], the ring Zp of integers modulo a prime p, and the ring M2 (Z) of 2 × 2 matrices over the integers are not integral domains.
What makes integral domains particularly appealing is that they have an important multiplicative group theoretic property, in spite of the fact that the nonzero elements need not form a group under multiplication. This property is cancellation. In an integral domain, if a ≠ 0 and ab = ac, then b = c.
Many authors prefer to define integral domains by the cancellation property—that is, as commutative rings with unity in which the cancellation property holds. This definition is equivalent to the one previously stated.
A field is a commutative ring with unity in which every nonzero element is a unit. It is easy to verify that every field is an integral domain, as if a and b belong to a field with a ≠ 0 and ab = 0, we can multiply both sides of the last expression by a^-1 to obtain b = 0. It is often helpful to think of ab^-1 as a divided by b. With this in mind, a field can be thought of as simply an integral domain in which division is possible.