IJPAM: Volume 18, No. 2 (2005)

VALUATION ON A RING WITH RESPECT
TO A SUBGROUP OF ITS GROUP OF UNITS

Angeliki Kontolatou$^1$, John Stabakis$^2$
$^{1,2}$Department of Mathematics
Faculty of Sciences
University of Patras
Patras, 26500, GREECE
$^1$e-mail: [email protected]
$^2$e-mail: [email protected]


Abstract.Given an integral domain $R$, $K$ its quotient field, $K^{*}$ the multiplicative group of $K, R^{*}$ the semi-group of non-zero elements of $R$ and $U(R)$ the multiplicative group of units of $R$, the canonical map of $K^{*}$ onto $K^{*}/U(R)$ is the well known semi-valuation. In this paper we prove that if - instead of $U(R)$ - we consider an adequate subgroup of $U(R)$, we may define another kind of valuation, the $G$-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 $G$-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