Exercise 14, § 7

Let $A$ be a local ring, $m$ its maximal ideal, and $q$ an ideal of $A$, distinct from $A$, such that $A/q$ is of finite length. Suppose that $A$ contains a subfield $k$ such that $A = k + m$. a) Now prove that there exist elements $x_1, ..., x_d$ of $q$, where $d = \dim(A)$, with $x_i \in q^{\delta_i}$, such that, for every sufficiently large integer v, one has $q^v = x_1.q^{v-\delta_1} + \cdots + x_d.q^{v-\delta_d}$. b) If one supposes the field A/m infinite, one can find an ideal a generated by d elements of q; where $d = \dim(A)$, such that $\overline{a} = \overline{q}$. (One may use § 7, no 5.)