Let R be a domain with fraction field L, and assume a is algebraic over L, show that is a nonzero ideal in R.

Results 1 to 4 of 4

- Oct 22nd 2008, 10:38 PM #1

- Joined
- Oct 2008
- Posts
- 147

- Oct 22nd 2008, 11:01 PM #2

- Joined
- May 2008
- Posts
- 2,295
- Thanks
- 7

- Oct 22nd 2008, 11:05 PM #3

- Joined
- Oct 2008
- Posts
- 147

- Oct 23rd 2008, 12:19 AM #4

- Joined
- May 2008
- Posts
- 2,295
- Thanks
- 7

you again repeated what you already posted! you should be more careful with posting your question! the correct and complete version of your question and the solution:

__Problem__: let be a domain and its field of fractions. show that if is algebraic over , then the set is a non-zero ideal of .

__Proof__: is obviously closed under addition because if then are integral over i.e. are in the integral closure of in since is a ring, we have

thus now suppose and we must show that : so and thus i.e. this proves that is an ideal

of the only thing left is to show that : since is algebraic over , we will have: for some integer and hence

so is integral over i.e.

__something for you to think about__:

did we use the assumption that is a domain and is its fraction field? is the claim in the problem true for any (unitary) commutative ring R and any ring extension L of R?