TITLE: Generalizing norms for modules
QUESTION [5 upvotes]: An normed space is defined as a vector space V plus a norm operation over $V$.
Is it meaningful to generalize this notion to modules, where one is dealing with rings instead of fields?
What I'm actually interested is to find the shortest non-zero "vector" over a subspace of, say $\mathbb{Z}_6^3$. As the addition and scalar multiplication are performed over a ring rather than a field, is defining a Euclidean norm $\|v\|_2=\sqrt{v\cdot v}$ meaningful?
PS: I'm actually asking this to better understand a comment raised by Daniel Lichtblau on another question of mine on Mathematica.SE.

REPLY [5 votes]: If you have a normed ring, then it makes good sense to study normed modules over the normed ring (with some compatibility between the norms on the ring and on the module).  This is a standard consideration in those parts of functional analysis that are closer to algebra (and in particular, in non-archimedean functional analysis).
On a finite field, the only sensible norm is the discrete norm, defined by $|a| = 0$ if $a = 0$ and $|a| = 1$ otherwise.  But this becomes slightly more interesting in your case, because $\mathbb Z_6$ is not a field. Rather, by the Chinese remainder theorem, we have $\mathbb Z_6 = \mathbb Z_2 \times \mathbb Z_3$, the product of two finite fields, and so we can give it a kind of product norm.  Actually, we have two choices, and $l_1$ version and an $l_{\infty}$ version.
In the former, we add up the norms on the components; in the latter, we take the max. of the norms of the two components.
So concretely, for the $l_1$ version, if $x = (a,b) \in \mathbb Z_2 \times \mathbb Z_3$, we define $|x| = 0$ if $a = b = 0$, $|x| = 1$ if $a$ or $b = 0$, but not both, and $|x| = 2$ if neither $a$ nor $b = 0$.
In the $l_{\infty}$ version, $|x| = 0$ if $x = 0$ and $|x | = 1$ otherwise.
The $l_{1}$ version looks more interesting to me, since it is more sensitive to the product structure.
You could now extend this to $\mathbb Z_6^3$, again in an $l_1$-way, i.e.
$|(x_1,x_2,x_3)| = |x_1| + |x_2| + |x_3|$.
Basically, this measures how different $(x_1,x_2,x_3)$ is from zero, by measuring how far each component $x_i$ is from zero, both mod $2$ and mod $3$.
I didn't look at the mathematica post you link in your question, so I don't know if this will help you.