Exercise 10, § 3

a) Two semi-absolute values $w_1, w_2$ on a ring $A$ are said to be equivalent if the pseudometrics $w_1(x - y), w_2(x - y)$ are equivalent. Show that, if $w$ is a semi-absolute value on $A$, then $aw$ and $w^{1/a}$ are semi-absolute values equivalent to $w$ for every real number $a \geq 1$.

b) If $w_i$ ($1 \leq i \leq n$) are semi-absolute values on a ring $A$, the functions $w = \sum_i w_i$ and $w' = \sup_i w_i$ are two equivalent semi-absolute values on $A$. If

then $a = \bigcap_i a_i$. If $A_i$ denotes the completion of the quotient ring $A/a_i$ endowed with the Hausdorff semi-absolute value associated with $w_i$, show that $A/a$ endowed with the Hausdorff semi-absolute value associated with $w$ is isomorphic to a subring of the product ring $\prod_i A_i$; show that (assuming that $A$ has an identity element 1) this ring is dense in $\prod_i A_i$ if and only if the $w_i$ satisfy the following condition: for each $\varepsilon > 0$ and for each index $i$, there is an element $x_i \in A$ such that $w_i(1 - x_i) \leq \varepsilon$ and $w_k(x_i) \leq \varepsilon$ for each index $k \neq i$.