Forward to §5.3 | Back to §5.1 | Up | Table of Contents | About this document

**Definition 5.2.1.**
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 in 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
R S.

An isomorphism from the commutative ring R onto itself is called an
**automorphism**
of R.

**(a)**The inverse of a ring isomorphism is a ring isomorphism.**(b)**The composition of two ring isomorphisms is a ring isomorphism.

**Proposition 5.2.3.**
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.

**Definition 5.2.4.**
Let : R -> S be a ring homomorphism.
The set

{ a in R | (a) = 0 }

is called the

**Proposition 5.2.5.**
Let : R -> S be a ring homomorphism.

**(a)**If a,b belong to ker() and r is any element of 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 in R and
(x) = s, defined by

(a_{0} +
a_{1}x
+ ... +
a_{m}x^{m}) =
(a_{0}) +
(a_{1})s
+ ... +
(a_{m})s^{m}.

**Theorem 5.2.6.
[Fundamental Homomorphism Theorem for Rings]**
Let : R -> S be a ring homomorphism.
Then

R / ker() (R).

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

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

**Example** 5.2.10.
The ring **Z**_{n}
is isomorphic to the direct sum of the rings
**Z**_{k}
that arise in the prime factorization of n.
This describes the structure of
**Z**_{n}
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
**Z**_{n}
and provides an easy proof of our earlier formula
for the Euler phi-function in terms of the prime factors of n.

**Definition 5.2.9.**
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**.

**Proposition 5.2.10.**
An integral domain has characteristic 0 or p, for some prime number p.

Forward to §5.3 | Back to §5.1 | Up | Table of Contents