Valuation on filtered module and relations

TL;DR

This paper introduces a valuation framework for filtered modules over filtered rings, establishing the existence of valuation skeletons and exploring their properties to deepen understanding of module structures.

Contribution

It develops a method to define valuations on filtered modules and constructs valuation skeletons, providing new tools for analyzing filtered module properties.

Findings

Existence of valuations on filtered modules

Construction of valuation skeletons for filtered modules

Properties of valuations derived from filtered structures

Abstract

In this paper we show if R is a filtered ring and M a filtered R module then we can define a valuation on a module for M. Then we show that we can find an skeleton of valuation on M, and we prove some properties such that derived form it for a filtered module.