Let A be a local ring with maximal ideal m. Let f E A, and define A(f) to be the multiplicity of the A-module AlAf with respect to m. Under suitable conditions I.A(fg) = AA(f) + iA(g). The relationship of ;iA to reduction of A, normalization of A and a quadratic transform of A is studied. It is then shown that there are positive integers ni,..., n. and rank one discrete valuations v1l..., vs of A centered at m such that pA(f) = n1v1(f) + * + n,v,(f) for all regular elements f of A. Let A be a nonnull noetherian local ring with maximal ideal m. Let d be the (Krull) dimension of A, the maximal length of a chain of prime ideals of A, excluding A. Let k be the residue field A/m, and let GmA be the associated graded ring of A with respect to m. Let f E A. If A/Af is of dimension d 1 define pA(f) to be em(A/Af), the multiplicity of the A-module A/Af relative to m in dimension d 1 [6, p. V-2] or the multiplicity of the local ring A/Af ([7, p. 2941, or[3, p. 75]). If A/Af is of dimension d, defne lA (f ) to be oo. Call 1A (f) the multiplicity of f (at m in A). If A is a regular local ring, PA is known to be the order valuation of A [3, 40.2, p. 154]. If A is entire ,A(fg) = ,A(f) + ,A(g) (Proposition 1, ?1). The order function VA of A [7, p. 2491 satisfies vA(f + g) > min {vA (f), v (g)}, and (Proposition 2, ? 1) VA is a valuation if and only if pA is a multiple of VA. If the ideal (0) is unmixed in A, pA is found to extend to the components of A (Lemma 2, ?2). If A is of dimension one, MA is found to extend to the normalization of A (Lemma 3, ?2). The extension of A to the first neighborhood ring of A (a quadratic transform of A) is found to preserve '1A (Lemma 4, ?3). This is used to prove the theorem of ?4, that there are positive integers nl, ... , ns and discrete rank one valuations vl, v.. V of A centered at m such that for every regular element f of A mA (f) = nv, (f) + * * * + n.,(f) Received by the editors September 19, 1974. AMS (MOS) subject classifications (1970). Primary 13H15; Secondary 13B20, 14B05. Copyright i) 1976, American Mathematical Society 321 This content downloaded from 157.55.39.163 on Wed, 21 Sep 2016 05:08:58 UTC All use subject to http://about.jstor.org/terms