Answer Happy

Let S be a domain whose elements are integral over a subring RC S. Show that if R is a field, so is S. (Hint: pick non-zero s ES and consider a monic relation of minimal degree.) Following is some useful hints (Only need prove above, do not need prove below) Definition 29.1. Let R be a commutative ring and let S be an R-algebra. (a) We say that S is module finite over R if it is a finitely generated R-module. (b) If RC S, we say that an element s ES is integral over R if there exists a monic polynomial f(X) = Xª + ad-₁Xd-¹ +...+ a₁X + ao € R[X] such that f(s) = 0. Proposition 29.2. Let R be a commutative ring. Let S be an R algebra and T be an S algebra. If S is module finite over R and T is module finite over S then T is an R-algebra which is module finite over R. Theorem 29.3. Let R be a commutative ring and let S be an R-algebra containing R. The following are equivalent: (a) S is module finite over R, - (b) there exist 01,..., On ES which are integral over R such that S = R[0₁,...,0n], (c) S is a finitely generated R-algebra and every element in S is integral over R.