# Hilbert - Samuel multiplicity of $B$ when there is a surjection $A \rightarrow B$

The setting is: Let $$A, B$$ be commutative, Noetherian, local rings, $$\phi:A \rightarrow B$$ a surjective homomorphism. Both rings also come with surjections $$\lambda_A, \lambda_B$$ to a DVR $$\mathcal{O}$$ which factor as $$\lambda_A = \lambda_B \circ \phi$$ (if this is helpful at all). Let $$e(A)$$ denote the Hilbert - Samuel multiplicity of $$A$$ w.r.t. to its maximal ideal. Considering $$B$$ as an $$A$$-module, I'm reading that $$e_A(B) \leq e_A$$ always holds, but I'm struggling to see why this is true. I went through the related chapters in Matsumura and Bruns - Herzog, but wasn't able to find an explanation, can anyone shed light on this?

There is a short exact sequence $$0 \to \ker \phi \to A \xrightarrow{\phi} B \to 0.$$ Hilbert-Samuel multiplicity is additive across short exact sequences (see Corollary 4.7.7 in Bruns and Herzog), so $$e(A)=e_A(B)+e_A(\ker \phi)$$, which proves the claim.