Exercise 9, § 3

a) On a ring A, two absolute semi-values $w_1, w_2$ are said to be equivalent if the distances $w_1(x - y), w_2(x - y)$ are equivalent. Now prove that, if $w$ is an absolute semi-value on A, $aw$ and $w^{1/a}$ are absolute semi-values equivalent to $w$ for every number $a \geq 1$.

b) If $w_i (1 \leq i \leq n)$ are absolute semi-values on a ring A, the functions $w = \sum_i w_i$ and $w' = \sup_i w_i$ are two absolute semi-values equivalent on A. If $a_i = w_i^{-1}(0)$ and $a = w(0)$, we have $a = \bigcap_i a_i$. If $A_i$ denotes the completion of the quotient ring $A/a_i$ endowed with the separated absolute semi-value associated with $w_i$, now prove that $A/a$, endowed with the separated absolute semi-value associated with $w$, is isomorphic to a subring of the product ring $\prod_i A_i$; for this subring to be everywhere dense in $\prod_i A_i$, it is necessary and sufficient (assuming that A has a unit element 1) that the absolute semi-values $w_i$ satisfy the following condition: for every $\varepsilon > 0$, and for each of the indices $i$, there exists an element $x \in A$ such that $w_i(1 - x) \leq \varepsilon$ and $w_k(x) \leq \varepsilon$ for every index $k \neq i$.