The definition of norm defined satisfies positive homogenity, triangle inequality, and separation of points.

Now, suppose, I have a module $M$ over an ordered ring $R$ where $R$ is endowed with an absolute value and is an ordered ring. Technically, I can still define a notion of norm on $M$ as a function from $M$ to $R$ satisfying the above properties (or replacing the triangle inequality by the ultra-metric one depending on the absolute value).

I was wondering, if such a general notion has been studied and if so, can someone point me to some references.


  • 1
    $\begingroup$ Hi - I'm a bit confused since your ring has both an absolute value and an ordering? Do they need to be compatible? $\endgroup$
    – John M
    Jul 25, 2011 at 1:33


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