A note on joint reductions and mixed multiplicities

Let $(A,\mathfrak{m})$ be a noetherian local ring with maximal ideal $\mathfrak{m}$ and infinite residue field $k=A\mathrm{/}\mathfrak{m}\mathrm{.}$ Let $J$ be an $\mathfrak{m}$-primary ideal, $I_1,\mathrm{.}\mathrm{.}\mathrm{.},I_s$ ideals of $A$, and $M$ a finitely generated $A$-module. In this paper, we interpret mixed multiplicities of $(I_1,\mathrm{.}\mathrm{.}\mathrm{.},I_s,J)$ with respect to $M$ as multiplicities of joint reductions of them. This generalizes the Rees’s theorem on mixed multiplicity (Theorem 2.4). As an application we show that mixed multiplicities are also multiplicities of Rees’s superficial sequences.