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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 }

 
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 + ... + a_mx^m) = (a₀) + (a₁)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.

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