Let R = {6 la, be Q}and let p:R – Q be defined by : 0( D = 1) is a ring a) Homomorphism. b) Isomorphism c) None 2) ker (4) is generated by ro a) 8 1 b) I J 3) R/ker (4) = a) Z b) Q c) R d) None 4) Ker (4) is: a) Prime ideal not maximal b) Maximal ideal not prime Both maximal and prime d) None

18. Find subrings and of such that is not a subring of .

29. Let be the set of Gaussian integers . Let . a. Prove or disprove that is a substring of . b. Prove or disprove that is an ideal of .

Prove that every ideal of n is a principal ideal. (Hint: See corollary 3.27.)

Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)

34. If is an ideal of prove that the set is an ideal of . The set is called the annihilator of the ideal . Note the difference between and (of Exercise 24), where is the annihilator of an ideal and is the annihilator of an element of.

Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .

. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .

Find the principal ideal (z) of Z such that each of the following sums as defined in Exercise 8 is equal to (z). (2)+(3) b. (4)+(6) c. (5)+(10) d. (a)+(b) If I1 and I2 are two ideals of the ring R, prove that the set I1+I2=x+yxI1,yI2 is an ideal of R that contains each of I1 and I2. The ideal I1+I2 is called the sum of ideals of I1 and I2.

In the ring of integers, prove that every subring is an ideal.