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

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§ 5.3 Ideals and Factor Rings

 
Definition 5.3.1 Let R be a commutative ring. A nonempty subset I of R is called an ideal of R if

(i) a ± b belongs to I for all a,b in I, and

(ii) ra belongs to I, for all a in I and r in R.

 
Proposition 5.3.2. Let R be a commutative ring with identity. Then R is a field if and only if it has no proper nontrivial ideals.

 
Definition 5.3.3. Let R be a commutative ring with identity, and let a be an element of R. The ideal

Ra = { x in R | x = ra   for some   r in R }

is called the principal ideal generated by a.
    An integral domain in which every ideal is a principal ideal is called a principal ideal domain.

 
Example 5.3.1. (Z is a principal ideal domain) Theorem 1.1.4 shows that the ring of integers Z is a principal ideal domain. Moreover, given any nonzero ideal I of Z, the smallest positive integer in I is a generator for the ideal.

 
For an ideal I of a commutative ring R, the set { a+I | a in R } of cosets of I in R (under addition) is denoted by R/I. By Theorem 3.8.4, the set forms a group under addition. The next theorem justifies calling R/I the factor ring of R modulo I.

 
Theorem 5.3.6. If I is an ideal of the commutative ring R, then R/I is a commutative ring, under the operations

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

for all a,b in R.

 
Proposition 5.3.7. Let I be an ideal of the commutative ring R.

(a) The natural projection mapping : R -> R/I defined by (a) = a+I for all a in R is a ring homomorphism, and ker() = I.

(b) There is a one-to-one correspondence between the ideals of R/I and the ideals of R that contain I.

 
Definition 5.3.8. Let I be a proper ideal of the commutative ring R. Then I is said to be a prime ideal of R if for all a,b in R it is true that if ab is in I then a is in I or b is in I.
    The ideal I is said to be a maximal ideal of R if for all ideals J of R such that I J R, either J = I or J = R.

 
Proposition 5.3.9. Let I be a proper ideal of the commutative ring R with identity.

(a) The factor ring R/I is a field if and only if I is a maximal ideal of R.

(b) The factor ring R/I is a integral domain if and only if I is a prime ideal of R.

(c) If I is maximal, then it is a prime ideal.

 
Theorem 5.3.10. Every nonzero prime ideal of a principal ideal domain is maximal.

 
Example 5.3.7. (Ideals of F[x]) Let F be any field. Then F[x] is a principal ideal domain, since by Theorem 4.2.2 the ideals of F[x] have the form I = <f(x)>, where f(x) is the unique monic polynomial of minimal degree in the ideal. The ideal I is prime (and hence maximal) if and only if f(x) is irreducible. If p(x) is irreducible, then the factor ring F[x]/<p(x)> is a field.

 
Example 5.3.8. (Evaluation mapping) Let F be a subfield of E, and for any element u in E define the evaluation mapping u : F[x] -> E by u(g(x)) = g(u), for all g(x) in F[x]. Since u(F[x]) is a subring of E that contains 1, it is an integral domain, and so the kernel of u is a prime ideal. Thus if the kernel is nonzero, then it is a maximal ideal, so F[x]/ker(u) is a field, and the image of u is a subfield of E.

 


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