We cannot use Theorem 10.15 directly on a field to obtain any new quotient fields, because the only ideals of a field are the zero ideal and the entire field. In fact, the following result shows that a field can be characterized by its ideals.

Theorem 10.22. The nontrivial commutative ring R is a field if and only if (0) and R are its only ideals.

Proof. Let I be an ideal in the field R. Suppose that I = (0), so that there is a nonzero element a I . Since a−1 R, a · a−1 = 1 I . Therefore, by Proposition 10.4, I = R. Hence R has only trivial ideals.

Conversely, suppose that (0) and R are the only ideals in the ring R. Let a be a nonzero element of R and consider (a) the principal ideal generated by a. Since 1 · a (a), (a) = (0), and hence (a) = R. Hence 1 R = (a), so

EXERCISES

For Exercises 10.1 to 10.6, find all the ideals in the rings.

For Exercises 10.7 to 10.10, construct addition and multiplication tables for the rings. Find all the zero divisors in each ring. Which of these rings are fields?

For Exercises 10.11 to 10.14, compute the sum and product of the elements in the given quotient rings.

10.11.3x + 4 and 5x − 2 in Q[x]/(x2 − 7).

10.12.x2 + 3x + 1 and −2x2 + 4 in Q[x]/(x3 + 2).

10.13.x2 + 1 and x + 1 in Z2[x]/(x3 + x + 1).

10.14.ax + b and cx + d in R[x]/(x2 + 1), where a, b, c, d R.

10.15. If U and V are ideals in a ring R, prove that U ∩ V is also an ideal in

R.

10.16.Show, by example, that if U and V are ideals in a ring R, then U V is

not necessarily an ideal in R. But prove that U + V = {u + v|u U, v V } is always an ideal in R.

10.17.Find a generator of the following ideals in the given ring and prove a general result for the intersection of two ideals in a principal ideal ring.

10.18. Find a generator of the following ideals in the given ring and prove a general result for the sum of two ideals in a principal ideal ring.

(a) (2) + (3) in Z. (b) (9) + (12) in Z.

(c) (x2 + x + 1) + (x2 + 1) in Z2[x].

10.19. (Second isomorphism theorem for rings) If I and J are ideals of the ring R, prove that

10.20.(Third isomorphism theorem for rings) Let I and J be two ideals of the ring R, with J I . Prove that I /J is an ideal of R/J and that

in these boolean rings are symmetric difference and intersection.

10.30. Let I be the set of all polynomials with no constant term in R[x, y]. Find a ring morphism from R[x, y] to R whose kernel is the ideal I . Prove that I is not a principal ideal.

10.31.Let I = {p(x) Z[x]| 5|p(0)}. Prove that I is an ideal of Z[x] by finding a ring morphism from Z[x] to Z5 with kernel I . Prove that I is not a principal ideal.

10.32. Let I P(X) with the property that, if A I , then all the subsets of A are in I , and also if A and B are disjoint sets in I , then A B I . Prove

10.36. Let a, b be elements of a euclidean ring R. Prove that

(a) (b) if and only if b|a.

For the rings in Exercises 10.37 and 10.38, find all the ideals and draw the poset diagrams of the ideals under inclusion.

Which of the elements in Exercises 10.39 to 10.46 are irreducible in the given ring? If an element is irreducible, find the corresponding quotient field modulo the ideal generated by that element.

10.56. Mn(R).

10.57. An ideal I = R is said to be a maximal ideal in the commutative ring R if, whenever U is an ideal of R such that I U R, then U = I or U = R. Show that the nonzero ideal (a) of a euclidean ring R is maximal if and only if a is irreducible in R.

10.58.If I is an ideal in a commutative ring R, prove that R/I is a field if and only if I is a maximal ideal of R.

10.59.Find all the ideals in the ring of formal power series, R[[x]]. Which of the ideals are maximal?

10.60.Let C[0, 1] = {f : [0, 1] → R|f is continuous}, the ring of real-valued

continuous functions on the interval [0, 1]. Prove that Ia = {f C[0, 1]|f (a) = 0} is a maximal ideal in C[0, 1] for each a [0, 1]. (Every maximal ideal is, in fact, of this form, but this is much harder to prove.)

10.61.If P is an ideal in a commutative ring R, show that R/P is an integral domain if and only if ab P can only happen if a P or b P . The ideal P is called a prime ideal in this case.

the radical of R.

(a)Show that N (R) is an ideal of R (the binomial theorem is useful).

(b)Show that N [R/N (R)] = {N (R)}.

(c)Show that N (R) is contained in the intersection of all prime ideals of R (see Exercise 10.61). In fact, N (R) equals the intersection of all prime ideals of R.

10.63.A commutative ring R is called local if the set J (R) of all non-invertible elements forms an ideal of R.

(a)Show that every field is local as is Zpn for each prime p and n 1, but that Z6 is not local.

Q for each prime p.

(c)If R is local show that R/J (R) is a field.

(d)If R/N(R) is a field, show that R is local (if a is nilpotent, then 1 − a is invertible).

10.64.If R is a (possibly noncommutative) ring, an additive subgroup L of R is called a left ideal if rx L for all r R and x L. Show that the only left ideals of R are {0} and R if and only if every nonzero element of R has an inverse (then R is called a skew field).

10.65.If F is a field, show that R = M2(F ) has exactly two ideals: 0 and R. (Because of this, R is called a simple ring.) Conclude that Theorem 10.22 fails if the ring is not commutative.