Exercise 7, § 3

Let L be a field, A a subring of L and $K \subset L$ the field of fractions of A. a) Show that if L is a finitely generated A-module (II, p. 206) then necessarily $A = K$ (by considering a supplementary subspace of K in L, qua vector space over K, show that K is a finitely generated A-module, hence necessarily of the form $s^{-1}A$, where $s \in A$; finally show that s must be invertible in A). b) Show that if there exist a finite number of elements $x_j$ ($1 \leq j \leq n$) of L, algebraic over K, such that $L = A[x_1, \ldots, x_n]$, then there exists an element $b \neq 0$ in A such that $K = A[b^{-1}]$ (show that there exists $b \neq 0$ in A such that L is a finitely generated $A[b^{-1}]$-module). Show that either b is invertible (and $K = A$) or every non-zero ideal of A contains a principal ideal $Ab^m$. Converse. The ring $k[[X]]$ of formal power series over a field k is a ring of this type.