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**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 theAn integral domain in which every ideal is a principal ideal is called a

**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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