Exercise 24, § 1

Let A be a torsion-free commutative group written additively. Let A be a commutative ring, B a commutative A-algebra and $h : A \to B$ the ring homomorphism defining the algebra structure on B. Then $h^{(\Delta)} : A^{(\Delta)} \to B^{(\Delta)}$ (Algebra,

Chapter II, § 11, Exercise 1) is a (ring) homomorphism of the algebra $A^{(\Delta)}$ of the group $A$ with respect to $A$ to the algebra $B^{(\Delta)}$ of the group $A$ with respect to $B$. Let $P = \sum_{\alpha \in \Delta} b_\alpha \otimes X^\alpha$ be an element of $B^{(\Delta)}$. Show that for $P$ to be integral over $A^{(\Delta)}$, it is necessary and sufficient that each of the elements $b_\alpha \in B$ is integral over $A$. (Reduce it to the case where $A$ is finitely generated and then to the case $A = \mathbf{Z}$.) Show that if $A$ is an integral domain and integrally closed so is $A^{(\Delta)}$.