Tensor product of linear forms.

Question

Let $P$ be a module over a conmutative unitary ring $A$ and $\mathcal{J}^r(P)$ be the module of $r$-linear forms $P \times \cdots \times P \longrightarrow A.$ If $\alpha \in \mathcal{J}^r(P)$ and $\beta \in \mathcal{J}^s(P)$ are two multilinear forms, its tensor product is, by definition: $$ \alpha \otimes \beta (x_1,\dots,x_{r+s}) := \alpha(x_1,\dots,x_r) \beta(x_{r+1},\dots,x_{r+s}) $$ My question is the following: is always $\mathcal{J}^{r+s}(P)$ with this operation a tensorial product of $\mathcal{J}^r(P)$ and $\mathcal{J}^s(P)$? I know it is true for free modules, but I don't know if it can be generalized, for example, to reflexive modules.