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7 c2021, y² + y5, zº — y²]. 6 Let K be any field. Find all elements in S = K[x, y, z] which are integral over R = K[x20
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, - R[0₁,...,0], (b) there exist 01,..., On ES which are integral over R such that S (c) S is a finitely generated R-algebra and every element in S is integral over R. Proposition 29.4. Let R be a subring of the commutative ring S and assume that all elements in S are integral over R. If a E R is a unit in S then it is a unit in R. Consequently, if S is a field, so is R.