Note on multiplicity

ABSTRACT. Let (R, M) be a local ring with infinite residue field and / = (xi,...,Xd)R an ideal generated by a system of parameters. It is shown that the multiplicity of / equals the multiplicity of IT where and R = R/(0: x%),N\axge. Introduction. Let (R, M) be a local ring with infinite residue field. A device commonly employed in studying the multiplicity of an M-primary ideal is to go mod a superficial element. The effect is to reduce the dimension of the ring yet preserve the multiplicity. This technique is particularly useful in proving theorems about multiplicity by induction on the dimension of R. There are however, oc