Forward to §5.2 | Back to §4.4 | Up | Table of Contents | About this document

The examples to keep in mind are these:
the set of integers **Z**;
the set **Z**_{n} 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 equationsa + 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

a**·**1 = a
and
1**·**a = a.

In this case, 1 is called a

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

**Example** 5.1.1.
(**Z**_{n})
The rings **Z**_{n}
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 e^{2} = e.
An element a is said to be
**nilpotent**
if there exists a positive integer n with
a^{n} = 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.

Forward to §5.2 | Back to §4.4 | Up | Table of Contents