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29. In Z[x], the ring of polynomials with integer coefficients, let / = (f(x) E Z[x] If(0) = 0}. Prove that I = (x). (This exercise is re- ferred to in this chapter and in Chapter 15.) 30. Show that A = {(3x, y) I x, y E Z) is a maximal ideal of Z O Z. Generalize. What happens if 3x is replaced by 4x? Generalize. 31. Let R be the ring of continuous functions from R to R. Show that A = {fERIf(0) = 0} is a maximal ideal of R. 32. Let R = Zg Zgo. Find all maximal ideals of R, and for each maxi- mal ideal 1, identify the size of the field R/I. 33. How many elements are in Z[i]/(3 + i)? Give reasons for your answer . 34. In Z[x], the ring of polynomials with integer coefficients, let / = {f(x) E Z[x] If(0) = 0}. Prove that I is not a maximal ideal. 35. In Z O Z, let I = {(a, 0) | a E Z). Show that I is a prime ideal but not a maximal ideal. 36. Let R be a ring and let I be an ideal of R. Prove that the factor ring R/I is commutative if and only if rs - sr E I for all r and s in R. 37. In Z[x], let I = {f(x) E Z[x] If(0) is an even integer]. Prove that I = (x, 2). Is / a prime ideal of Z[x]? Is I a maximal ideal? How many elements does Z[x]/I have? (This exercise is referred to in this chapter.) 38. Prove that I = (2 + 2i) is not a prime ideal of Z[i]. How many elements are in Z[i]//? What is the characteristic of Z[i]/I? 39. In Z, [x], let / = (x2 + x + 2). Find the multiplicative inverse of 2x + 3 + / in Z,[x]/I. 40. Let R be a ring and let p be a fixed prime. Show that I, = {r ER I additive order of r is a power of p} is an ideal of R. 41. An integral domain D is called a principal ideal domain if every ideal of D has the form (a) = {ad l d E D} for some a in D. Show that Z is a principal ideal domain. (This exercise is referred to in Chapter 18.) 42. Let R = 1 8 2 ab.dez} ands - to " ]lostezs is even . If S is an ideal of R, what can you say about r and t