Let $R_1,\ldots, R_n$ be (commutative) rings, and $M_1 , \ldots, M_n$ be modules. What are the minimal assumpsions about the rings and modules needed in order to define a tensor product $M_1 \otimes_{R_1} M_2 \otimes _{R_2} \otimes\dots\otimes_{R_n} M_n$?
I am aware of an existing similar question. However, I am mainly looking for a reference that deals with a similar construction. In particular, I am looking for a proof of that the module structure ($(R_1,\ldots,R_n)$-multi-module structure?) is well-defined, and the universal property $$ \operatorname{Hom}_{(R_1, ..., R_n)}(M_1 \otimes_{R_1}\dots\otimes_{R_n} M_n,P) \cong \operatorname{Mult}_{(R_1, ..., R_n)}(M_1,\ldots,M_n; P)$$
Where $P$ is an Abelian group with a structure of a $R_i$-module for each $i$, and $\operatorname{Mult}_{(R_1, ..., R_n)}(M_1,\ldots,M_n; P)$ is the set of maps which are $R_i$-linear in the $i$'th coordinate.