Exercise 9, § 3

A mapping $w$ of a ring $A$ into $\mathbf{R}_+$ is called a semi-absolute value on $A$ if it satisfies the conditions : (i) $w(0) = 0$; (ii) $w(x-y) \leq w(x) + w(y)$;

(iii) $w(xy) \leq w(x)w(y)$. The semi-absolute value $w$ is said to be Hausdorff if $w(x) = 0$ implies $x = 0$. If $w$ is a Hausdorff semi-absolute value on $A$, then $w(x - y)$ is an invariant metric on the additive group of $A$, and hence defines a topology compatible with the additive group structure of $A$. Generalize the principal properties of normed algebras, notably Proposition 13, to rings endowed with a semi-absolute value.

If $w$ is a non-Hausdorff semi-absolute value on $A$, the set $\overline{w}(0)$ is a two-sided ideal $a$ in $A$; if we endow $A$ with the topology defined by the pseudometric $w(x - y)$, this topology is compatible with the ring structure of $A$, and the Hausdorff space associated with $A$ is the quotient ring $A/a$, on which the function $\overline{w}(\bar{x})$, which for each $\bar{x}$ mod $a$ is equal to the common value of $w(x)$ for all $x \in \bar{x}$, is a Hausdorff semi-absolute value said to be associated with $w$; the topology defined by $\overline{w}$ is the quotient by $a$ of the topology of $A$.