Math 514 homework #8 Due date: 19950920 D. J. Bernstein 19950913 Let A be a ring, R a subring of A. We say that α ∈ A is integral over R if α is a root of some monic polynomial with coefficients from R. The integral closure of R in A is the set of α ∈ A that are integral over R. A domain R is integrally closed if it equals its integral closure in its field of fractions; this means that if a/b is integral over R, with a ∈ R and 0 ≠ b ∈ R, then a/b ∈ R. Exercise 23. Let R be a domain. Assume that I ÷ I = R for every nonzero finitely generated ideal I of R. Prove that R is integrally closed. (Hint: If a/b is a root of the polynomial z^{d} + f_{d−1}z^{d−1} + ⋯ + f_{1}z + f_{0}, consider the ideal I = a^{d−1}R + a^{d−2}bR + ⋯ + ab^{d−2}R + b^{d−1}R.)