Prolongations Of A Valuation On A Module

In 2006, Zeng Guangxing and Yang Pu [7] introduced the notion of valuations on modules, and obtain many important results. In this paper, we investigate the prolongation of valuations on a module over a commutative ring. By a counterexample, it is illustrated that a valuation on a module need not be prolongable on an extension module. Based on some relevant facts about the prolongation of valuations on a module, we obtains two necessary and sufficient conditions for a valuation on a module to have a prolongation.This paper is divided into the three sections as follows:Section 1 is the introduction of this paper. In this section, we describe the current status of the research on valuations of modules.In section 2, we introduce the notion of prolongations of valuations in the category of modules over commutative rings, and establish some relevant results. As main results in this section, we obtain two necessary and sufficient conditions for a valuation on a module to have a prolongation.In section 3, we investigate the prolongation of Mains valuations with respect to a homomorphism and the prolongation of valuations on a module with respect to a pair of homomorphisms. As a main result in this section, we obtain a necessary and sufficient conditions for a valuation on a module to have a prolongation with respect to a given pair of homomorphisms.