Exercise 18, § 7

Let A be a ring and $a$ an ideal of A. For every element $a \in A$, denote by $v_a(a)$ the supremum of the set of integers $\nu \in \mathbf{N}$ such that $a \in a^\nu$. (a) Prove that if $a \notin \bigcap_{k=0}^\infty a^k$ the sequence $\left( \frac{v_a(a^k)}{k} \right)$ is convergent. Its limit will be denoted by $\bar{v}_a(a)$. If $a \in \bigcap_{k=0}^\infty a^k$ we put $\bar{v}_a(a) = +\infty$. (b) Prove that, for every $a \in A$, one has $\bar{v}_a(a) = \bar{v}_{\overline{a}}(a)$ and deduce that the following conditions are equivalent: (i) $\bar{v}_a(a) \geqslant 1$, (ii) $a \in a$. (Study the monotonicity of the sequence considered in (a).)