Mathematics 667
Homework (due Apr. 15)
A. Hulpke
32) Let I ⊲ A be an irreducible ideal (i.e. it cannot be expressed as intersection of two strictly larger
ideals). Show that the following properties are equivalent:
i) I is primary.
ii) For every multiplicatively closed subset S ⊂ A we have that (S −1 I)c = (I ∶ x) for some x ∈ S.
iii) For every x ∈ A the sequence (I ∶ x n ) becomes stationary.
33) Suppose that B is integral over A. Show:
a) Is a ∈ A is a unit in B, then a −1 ∈ A.
b) Jac(B)c = Jac(A) (where Jac denotes the Jacobson-radical).
34) Let A be a subring of B such that B − A is closed under multiplication. Show that A is integrally
closed in B.
35) Let K ≥ Q be a finite extension and A the integral closure of Z in K.
a) Show that A is a free Z module. This ring is called the ring of algebraic integers in K. (Hint):
Consider a Q basis of K, then A is a submodule of the Z-span of this basis.)
b) A free Z-generating system for A is called an integral basis of K. Show that an integral basis for
A is a Q-basis for K.
c) Let B 1 and B2 be two integral bases for K and M the matrix for the base change. Show that M ∈
Zn×n and has determinant ±1. (Such a matrix is called unimodular.)
36) Let K be a field (you may assume that it is algebraically closed) and K(x) the fraction field of
the polynomial ring K[x].
a) Show that K[x] is integrally closed in K(x).
b) Find the integral closure of K[x 2 , x(x 2 − 1)] in K(x).
c) Find a subring K[y] of A = K[x, x −1 ] such that A is a finitely generated K[y] module