3. a) Let J = (2) in the ring Z10. Define the set I as I = {re 10 | rt = 0 for every t = J}. Find the elements of I explicitly. b) Is the set / from part (a), an ideal? Justify your answer. c) Now, you need to prove the following result (in general). Prove: For any ideal / in a ring R, the set I described below is an ideal in R. I={r ER❘rt OR for every tЄ J}

Label each of the following statements as either true or false. The only ideals of the set of real numbers are the ideals of {0} and .

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.

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.)

15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .

. 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 .

18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .

Prove that if R is a field, then R has no nontrivial ideals.

11. Show that defined by is not a homomorphism.