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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.
5.1.1. Definition 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 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 R,

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

(iii) Commutative laws: For all a,b R,
a+b = b+a and a·b = b·a.

(iv) Distributive laws: For all a,b,c 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 R,
a+0 = a and 0+a = a.

(vi) Additive inverses: For each a R, the equations
a+x = 0 and x+a = 0

have a solution x 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 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.

5.1.2. Definition 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.

5.1.3. Proposition 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 R, then -a R.
5.1.4. Definition Let R be a commutative ring with identity element 1. An element a R is said to be invertible if there exists an element b 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.
5.1.5. Proposition 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.

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.

5.2.1. Definition Let R and S be commutative rings. A function :R->S is called a ring homomorphism if

(a+b) = (a) + (b) and (ab) = (a)(b)

for all a,b R.
A ring homomorphism that is one-to-one and onto is called an isomorphism. If there is an isomorphism from R onto S, we say that R is isomorphic to S, and write RS. An isomorphism from the commutative ring R onto itself is called an automorphism of R.

5.2.2. Proposition

(a) The inverse of a ring isomorphism is a ring isomorphism.

(b) The composition of two ring isomorphisms is a ring isomorphism.
5.2.3. Proposition Let :R->S be a ring homomorphism. Then
(a) (0) = 0;

(b) (-a) = -(a) for all a in R;

(c) if R has an identity 1, then (1) is idempotent;

(d) (R) is a subring of S.
5.2.4. Definition Let :R->S be a ring homomorphism. The set
{ a R | (a) = 0 }

is called the kernel of , denoted by ker().
5.2.5. Proposition Let :R->S be a ring homomorphism.

(a) If a,b ker() and r R, then a+b, a-b, and ra belong to ker().

(b) The homomorphism is an isomorphism if and only if ker() = {0} and (R) = S.
Example 5.2.5. Let R and S be commutative rings, let :R->S be a ring homomorphism, and let s be any element of S. Then there exists a unique ring homomorphism :R[x]->S such that
(r) = (r) for all r R and (x) = s, defined by
(a0 + a1x + ... + amxm) = (a0) + (a1)s + ... + (am)sm.

5.2.7. Proposition Let R and S be commutative rings. The set of ordered pairs (r,s) such that r R and s S is a commutative ring under componentwise addition and multiplication.

5.2.8. Definition Let R and S be commutative rings. The set of ordered pairs (r,s) such that r R and s S is called the direct sum of R and S.

Example 5.2.10. The ring Zn is isomorphic to the direct sum of the rings Zk that arise in the prime factorization of n. This describes the structure of Zn in terms of simpler rings, and is the first example of what is usually called a ``structure theorem.'' This structure theorem can be used to determine the invertible, idempotent, and nilpotent elements of Zn and provides an easy proof of our earlier formula for the Euler phi-function in terms of the prime factors of n.

5.2.9. Definition Let R be a commutative ring with identity. The smallest positive integer n such that (n)(1) = 0 is called the characteristic of R, denoted by char(R). If no such positive integer exists, then R is said to have characteristic zero.