Let $F\hspace{.02 in}$ be a field. $\:$ Let $E\hspace{.02 in}$ be a non-zero subring of $F$.
Let $\hspace{.03 in}\leq\hspace{.03 in}$ be a total order on $E\hspace{.02 in}$ that makes $E\hspace{.02 in}$ an ordered ring. $\;\;\;$ Let $\;\; |\cdot| \: : \: F\to E \;\;$ be a function.
I will require the following to hold for all members $x$ of $E$ and for all members $y$ and $z$ of $F\hspace{.03 in}$:
(a) $\qquad 0 \leq x \;\; \implies \;\; |x| = x$
(b) $\qquad 0 \leq |y|$
(c) $\qquad |y| = 0 \;\; \iff \;\; y = 0$
(d) $\qquad |y\cdot z| \;\; = \;\; |y| \cdot |z|$
(e) $\qquad |y+z| \;\; \leq \;\; |y| + |z|$
1. $\;$ Is there a name for what I've just set up?
Now, I'm wondering about the redudancy in those conditions. $\;$ I know that
neither (a) nor (b) follows from the other four, and that (c) follows from [(a) and (d)].
${}$2. $\;$ Does (d) follow from the other four?
3. $\;$ Does (e) follow from [(a) and (b) and (d)]?