Thread: Please help with abstract algebra

Im studying for my final next week and I need a little help.

"+ denotes an external direct product"

1. Find the number of elements of order 4 in Z_60+Z_12+Z_12.
2. In the ring Z+Z show: A = {(3x,y)|x,y in Z} is a maximal ideal.

For number 1 the LCM(60,12)=60 then is that the number of elements of order 4?

For number 2, I need some serious help starting.

Thanks Guys

Consider the fact that and .

Applying these relations we have the following:


It is clear that since 4 does not divide 3 or 5 there will be no elements of order 4 in or respectively (Lagrange). Furthermore, it is obvious that 1 and 3 have order 4 in . Thus any combination of these elements in the 3 copies of will work so long as you choose 0 in the rest of the places. Note obviously not ALL of the componants can be 0 or else it is the identity and has order 1. I claim that there will be elements, you oughta be able to prove why there are no more and no less.

Certainly is a Euclidean Domain which can be shown to be a Principal Ideal Domain. In a PID every nonzero prime ideal is maximal. It is pretty clear that the second coordinate is all of , so really this just amounts to showing that the first coordinate is maximal (I think anyway). The first coordinate is simply the principal ideal generated by (3) which is a prime ideal, so it is also maximal. Thus any ideal containing A would have to contain all of in the second coordinate, and would have to contain (3) in the first coordinate, but since (3) is maximal in this means any ideal satisfying this is necessarily all of or else (3) again, and so if B were an ideal containing A then either or and so A is maximal.

Not sure where exactly you got hung up on this one, but it is pretty easy to show both that E.D. implies PID, and then in a PID that prime ideals are maximal, just follow definitions, nothing too tricky about that, plus almost any algebra book would have proofs of these facts I should think.