Prime Ideals in Z[sqrt(2)] and Cosets in ZxZ/I

[kobulingam]

I'm in an algebra (ring) class and I'm looking at a previous midterm (I have attached it here to prove that this is not homework problem).

Can anyone tell me how to answer question 3 and 5? I will repeat again in here:

3) Let R = Z[sqrt(2)] and P = <sqrt(2)> the ideal generated by sqrt(2)

a) Prove that P is a prime ideal.
b) Let D = Rp (localization of R at complement of P, aka ring of fractions of R w.r.t. (R-P) )
Let Pp be ideal of Rp generated by sqrt(2). Is Pp a prime ideal of Rp? Prove answer.

5) Let R = ZxZ (direct product of integer sets with operations defined as usual componentwise). Let I = < (4,9), (6,12) > ideal generated by those two elements. How many elements (cosets of I) does R/I have?


Any help would be appreciated. Let me repeat that this is not homework, as I have attached file proving that these are past test questions (which I got from school's math society website).

 

[HallsofIvy]

For (3), first show how any member of Z[sqrt(2)] can be written. How is that related to the ideal <sqrt(2)>? IS that an ideal in Z[sqrt(2)]? What does it mean to say it is a prime ideal?

For 5, again, what IS an ideal of ZxZ? Can you show that the set generated by <(4,9), (6,12)> IS an ideal? What are "cosets" of that set?

 

[kobulingam]

For (3), first show how any member of Z[sqrt(2)] can be written. How is that related to the ideal <sqrt(2)>? IS that an ideal in Z[sqrt(2)]? What does it mean to say it is a prime ideal?

For 5, again, what IS an ideal of ZxZ? Can you show that the set generated by <(4,9), (6,12)> IS an ideal? What are "cosets" of that set?

I got 3)a) by doing exactly what you are implying, but can't get it to work for 3)b).


For 5), yes, a set generated by memebers of a ring like that are always ideals.