Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

## § 5.1 Commutative Rings, Integral Domains

The examples to keep in mind are these: the set of integers Z; the set Zn of integers modulo n; any field F (in particular the set Q of rational numbers and the set R of real numbers); the set F[x] of all polynomials with coefficients in a field F. The axioms are similar to those for a field, but the requirement that each nonzero element has a multiplicative inverse is dropped, in order to include integers and polynomials in the class of objects under study.

Definition 5.1.1 Let R be a set on which two binary operations are defined, called addition and multiplication, and denoted by + and ·. Then R is called a commutative ring with respect to these operations if the following properties hold:

(i) Closure: If a,b belong to R, then the sum a+b and the product a·b are uniquely defined and belong to R.

(ii) Associative laws: For all a,b,c in R,

a+(b+c) = (a+b)+c     and     a·(b·c) = (a·b)·c.

(iii) Commutative laws: For all a,b in R,

a+b = b+a     and     a·b = b·a.

(iv) Distributive laws: For all a,b,c in R,

a·(b+c) = a·b + a·c     and     (a+b)·c = a·c + b·c.

(v) Additive identity: The set R contains an additive identity element, denoted by 0, such that for all a in R,

a + 0 = a     and     0 + a = a.

(vi) Additive inverses: For each a in R, the equations

a + x = 0     and     x + a = 0

have a solution x in R, called the additive inverse of a, and denoted by -a.

The commutative ring R is called a commutative ring with identity if it contains an element 1, assumed to be different from 0, such that for all a in R,

a·1 = a     and     1·a = a.

In this case, 1 is called a multiplicative identity element or, more generally, simply an identity element.

As with groups, we will use juxtaposition to indicate multiplication, so that we will write ab instead of a·b.

Example 5.1.1. (Zn) The rings Zn form a class of commutative rings that is a good source of examples and counterexamples.

Definition 5.1.2 Let S be a commutative ring. A nonempty subset R of S is called a subring of S if it is a commutative ring under the addition and multiplication of S.

Proposition 5.1.3. Let S be a commutative ring, and let R be a nonempty subset of S. Then R is a subring of S if and only if

(i) R is closed under addition and multiplication; and

(ii) if a belongs to R, then -a belongs to R.

Definition 5.1.4. Let R be a commutative ring with identity element 1. An element a in R is said to be invertible if there exists an element b in R such that ab = 1.
The element a is also called a unit of R, and its multiplicative inverse is usually denoted by a-1.

An element e of a commutative ring R is said to be idempotent if e2 = e. An element a is said to be nilpotent if there exists a positive integer n with an = 0.

Proposition 5.1.5. Let R be a commutative ring with identity. Then the set R× of units of R is an abelian group under the multiplication of R.

Definition 5.1.6. A commutative ring R with identity is called an integral domain if for all a,b in R, ab = 0 implies a = 0 or b = 0.

Theorem 5.1.7 Let F be a field with identity 1. Any subring of F that contains 1 is an integral domain.

Theorem 5.1.8. Any finite integral domain must be a field.

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