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

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    Algebra Midterm
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SUMMARY

The discussion focuses on two algebra problems involving prime ideals and cosets. In question 3, the ideal P = <sqrt(2)> in the ring R = Z[sqrt(2)] is analyzed to determine if it is a prime ideal, with localization D = Rp being introduced for further exploration. Question 5 examines the ideal I = <(4,9), (6,12)> in the direct product R = ZxZ, seeking to find the number of cosets in R/I. Key insights include the necessity of demonstrating that P is a prime ideal and understanding the structure of ideals and cosets in these algebraic systems.

PREREQUISITES
  • Understanding of ring theory and ideals, specifically in the context of Z[sqrt(2)]
  • Familiarity with localization of rings, particularly Rp in relation to prime ideals
  • Knowledge of direct products of rings, specifically ZxZ and its operations
  • Concept of cosets and their relation to ideals in algebraic structures
NEXT STEPS
  • Study the properties of prime ideals in commutative rings, focusing on examples like Z[sqrt(2)]
  • Learn about localization techniques in ring theory, particularly the construction of Rp
  • Investigate the structure of ideals in direct products of rings, using ZxZ as a case study
  • Explore the concept of cosets in algebra, including their applications in quotient rings
USEFUL FOR

Students of abstract algebra, particularly those studying ring theory, as well as educators and mathematicians interested in the properties of ideals and cosets in algebraic structures.

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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).
 

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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?
 
HallsofIvy said:
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.
 

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