Decomposition Of M-Valuations On A Commutative Ring

In 2002,DegenZhang introduced the notion of M-valuations on commutative rings.Both Manis valuations and formally finite V-valuations may be considered as special examples.The purpose of this paper is to investigate the decomposition and composition of M-valuations on a commutative ring.By introducing the notion of harmonic congruences for ordered monoids,a M-valuation on a commutative ring is decomposed into a cancellative M-valuation and a Manis valuation with null core on the residue ring of this cancellative M-valuation.Conversely,for a cancellative M-valuation and a Manis valuation with null core on the residue ring of this cancellative M-valuation,a M-valuation is so composed that it can be turned back into the given valuations by such a decomposition.