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