# IJPAM: Volume 18, No. 2 (2005)

**VALUATION ON A RING WITH RESPECT**

TO A SUBGROUP OF ITS GROUP OF UNITS

TO A SUBGROUP OF ITS GROUP OF UNITS

Department of Mathematics

Faculty of Sciences

University of Patras

Patras, 26500, GREECE

e-mail: kontolat@math.upatras.gr

e-mail: jns@math.upatras.gr

**Abstract.**Given an integral domain ,
its quotient field, the multiplicative group of
the semi-group of non-zero elements of and the
multiplicative group of units of , the canonical map of
onto is the well known semi-valuation. In this paper
we prove that if - instead of - we consider an adequate
subgroup of , we may define another kind of valuation, the
-valuation, whose the value group has torsion and the triangle
property differs slightly from the one of the semi-valuation. We
prove that both of these triangle properties may be presented by
two completions of an ordered space. Apart of some direct
algebraic and topological consequences of the new definition we
construct a -valuated field with value group a given
splitting Abelian ordered group.

**Received: **December 1, 2004

**AMS Subject Classification: **12J25, 13A18, 54E15, 06F15

**Key Words and Phrases: **generalized valuations, completions of ordered spaces, uniformity and proximity

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2005

**Volume:** 18

**Issue:** 2