Exercise 8, § 3

One says that a mapping $w$ of a ring A into $\mathbf{R}_+$ is a semi-absolute value on A if it satisfies the conditions 1: 1°: $w(0) = 0$; 2° $w(x - y) \leq w(x) + w(y)$; 3° $w(xy) \leq w(x)w(y)$. The semi-absolute value $w$ is said to be separated if $w(x) = 0$ implies $x = 0$. If $w$ is a separated semi-absolute value on A, $w(x - y)$ is an invariant distance on the additive group A, and consequently defines a topology compatible with the additive group structure of A; now prove that this topology is compatible with the ring structure of A. Generalize to rings provided with a semi-absolute value the principal properties of normed algebras, in particular prop. 14 (IX, p. 40).

If $w$ is a non-separated semi-absolute value on A, the set $w(0)^{-1}$ is a two-sided ideal $a$ in A; if A is provided with the topology defined by the distance $w(x - y)$, it is compatible with the ring structure of A, and the associated separated space of A is none other than the quotient ring $A/a$, on which the function $\overline{w}(\bar{x})$, equal for every class $\bar{x}$ mod. $a$ to the common value of $w(x)$ for all the $x \in \bar{x}$, is a separated semi-absolute value, said to be associated with $w$, defining the quotient topology of that of A by $a$.