The Decomposition Of Valuations On A Module

The purpose of this paper is to investigate the decomposition of valuations on a module over a commutative ring.let M be a module over a commutative ring R,let v:Mâ†'â–³be a valuation on M,and let (?) be the value group induced byν. By introducing the notions of harmonic equivalences onâ–³andν-isolated subgroups of (?),the closed interplay of harmonic equivalenccs onâ–³andν-isolated subgroups of (?) is invstigated.A main result in this paper is established as follows:For aν-isolated subgroup∑of (?),νcan decomposed into a new valution on M and a Manis valution with null core and value group∑on the residue class ring induced by this new valution..