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Sorry about the formatting, LaTeX doesn't seem to be working, it seems to be giving garbage output.
Find a nontrivial proper ideal of Z x Z that is not prime.
Defn: An ideal N in a commutative ring R is prime, if ab in N implies, either a in N, or b in N, for all a, b in R.
Thm: An ideal of N is prime in R iff R/N is an integral domain.
The solution at the back of the book is 4Z x {0}, however I don't see how this can be true. If we apply the definition of a prime ideal, we can show this ideal is in fact prime.
The product of 2 typical elements of N = 4Z x {0} is (a,b)(c,d) = (ac, bd), with bd = 0. Since b and d are in Z, which is an integral domain, bd = 0 implies b = 0 or d = 0. So (a,b) or (b,d) is in N. Thus N is prime by definition.
Alternatively, we can compute (Z x Z) / (4Z x {0}) = { (a,b) + 4Z x {0} | (a,b) in Z x Z}. Since, by the division algorithm, we can write a = 4q + r for some r = 0, 1, 2, 3, this simplifies to: (Z x Z) / (4Z x {0}) = { (a,b) + 4Z x {0} | a in {0, 1, 2, 3}, b in Z}, which is clearly isomorphic to {0 ,1 , 2, 3} x Z, which is isomorphic to Z by the map f((a,b)) = 4*a + b, for (a,b) in {0 ,1 , 2, 3} x Z. But Z is a domain, so (Z x Z) / (4Z x {0}) is also a domain and by the theorem above, 4Z x {0} is prime.
So did I make a huge mistake somewhere, or is the solution 4Z x {0} wrong. If so, how do I find the correct solution?
Homework Statement
Find a nontrivial proper ideal of Z x Z that is not prime.
Homework Equations
Defn: An ideal N in a commutative ring R is prime, if ab in N implies, either a in N, or b in N, for all a, b in R.
Thm: An ideal of N is prime in R iff R/N is an integral domain.
The Attempt at a Solution
The solution at the back of the book is 4Z x {0}, however I don't see how this can be true. If we apply the definition of a prime ideal, we can show this ideal is in fact prime.
The product of 2 typical elements of N = 4Z x {0} is (a,b)(c,d) = (ac, bd), with bd = 0. Since b and d are in Z, which is an integral domain, bd = 0 implies b = 0 or d = 0. So (a,b) or (b,d) is in N. Thus N is prime by definition.
Alternatively, we can compute (Z x Z) / (4Z x {0}) = { (a,b) + 4Z x {0} | (a,b) in Z x Z}. Since, by the division algorithm, we can write a = 4q + r for some r = 0, 1, 2, 3, this simplifies to: (Z x Z) / (4Z x {0}) = { (a,b) + 4Z x {0} | a in {0, 1, 2, 3}, b in Z}, which is clearly isomorphic to {0 ,1 , 2, 3} x Z, which is isomorphic to Z by the map f((a,b)) = 4*a + b, for (a,b) in {0 ,1 , 2, 3} x Z. But Z is a domain, so (Z x Z) / (4Z x {0}) is also a domain and by the theorem above, 4Z x {0} is prime.
So did I make a huge mistake somewhere, or is the solution 4Z x {0} wrong. If so, how do I find the correct solution?