2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d) just ideal

(7) Are the following ideals. Prime, Maximal or neither in the rings R? Why? R = 26x12 (a) (x+5), (b) 22x+4> R=22 [²] R = Q[²] 224x521, R = 21 x 21 (e) 71 x 7 26, R = 21 x 21 (f) (c) , {a+b√3|319}, R = 2/(√3) Татов

6. (10%) Let R be the ring of R of all real numbers of the form x+yv2 with respect to standard addition and multiplication, where r and y are integers, and let T be the subset of T of the numbers of the form m + ny2, where m is an even integer, and n is an integer. Prove that T is an ideal of R.

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}