Multipliers of Von Neumann regular rings

We analyse the structure of the multiplier ring M(R) of a (nonunital) Von Neumann regular ring R. We show that M (R) is not regular in general, but every principal right ideal is generated by two idempotents. This, together with Riesz Decomposition on idempotents of M(R), furnishes a description of the monoid V(M(R)) of Murray-Von Neumann equivalence classes of idempotents which is used to effectively examine the lattice of ideals of M(R). The techniques developed here allow other applications to the category of projective modules over regular rings.