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

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§ 5.2 Ring Homomorphisms

 
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.

 
Proposition 5.2.2.

(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 kernel of , denoted by ker().

 
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

(a0 + a1x + ... + amxm) = (a0) + (a1)s + ... + (am)sm.

 
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 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.

 
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.

 


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