Tensor product generalization - Proving associativity $\require{mathtools}$
Let $R$ denote a commutative ring with unity. I tried to generalize the definitions of tensor product of a finite number of left $R$-modules to the tensor product of a set of left $R$-modules indexed by an arbitrary indexing set $I$. These are mentioned below. (Hereon, all modules are assumed to be left $R$-modules)
CONTEXT:
Let $(M_{i})_{i\in I}$ be a family of $R$-Modules.    Let $S$ be an $R-$module. Then, $\phi :\times_{i\in I}M_i\rightarrow S $ is called multilinear if:

$$((m_{i})_{i\in I},(s_{i})_{i\in I},(t_{i})_{i\in I}\in \times_{i\in I}M_{i} \text{ and } j\in I \text{ such that } m_{j}=s_{j}+rt_{j}\text{ and }\forall i\in I\setminus\{ j \},m_{i}=s_{i}=t_{i}) \Rightarrow (\phi((m_{i})_{i\in I})=r\phi((t_{i})_{i\in I})+\phi((s_{i})_{i\in I})) $$

An ordered pair $(T,t: \times_{i\in I} M_{i}\rightarrow T)$ of an $R$-module $T$ and an $R$-morphism $t$ is said to be the (upto isomorphism) tensor product of $\times_{i\in I} M_{i}$  if for every multilinear map $f: \times_{i\in I} M_{i}\rightarrow W$, there is a unique linear map $g:T\rightarrow W$ such that $f=g \circ t$. The respective $R$-module of the tensor product is denoted by $\bigotimes_{i\in I} M_{i}$.
ATTEMPTS, EFFORTS AND QUESTION: I tried to check whether some elementary properties of the finite index tensor product hold for this generalization. For instance, commutativity seems to hold. However, I'm not able to prove or disprove whether associativity holds. That is, if $\{I_p|p\in \mathcal{P}\}$ is a partition of $I$, then:

$$\bigotimes_{i\in I}M_{i} \cong \bigotimes_{p\in \mathcal{P}}\left(\bigotimes_{i\in I_{p}}M_{i}\right)$$

I'm not able to get a good grasp of the structure. I can't make out if it's true either. Any help in this direction is greatly appreciated. Thank you.
PS: I'm unfamiliar with category theory.
 A: This definition is unfortunately rather poorly behaved for infinite index sets, and in particular is not associative.  For instance, $R$ ought to be a "unit" for the tensor product, but a tensor product of infinitely many copies of $R$ is typically much larger than $R$.  Indeed, $\bigotimes M_i$ is can be constructed explicitly by starting with the free module on $\prod M_i$ then imposing relations for multilinearity.  Note, though, that two elements of $\prod M_i$ can be linked by a finite sequence of relations only if they agree on all but finitely many coordinates.  So, if you partition $\prod M_i$ into equivalence classes by the equivalence relation of being equal on all but finitely many coordinates, there are no relations between the different equivalence classes, and so $\bigotimes M_i$ ends up being a direct sum of the modules generated by each equivalence class.  Typically uncountably many of these direct summands will be nontrivial.  In particular, if $R$ is a field and every $M_i$ is $R$, each equivalence class where there are not infinitely many coordinates that are $0$ will just generate a copy of $R$ (for the same reason that a finite tensor product of copies of $R$ is $R$).  But if $I$ is infinite and $R$ has more than two elements, there are a huge number of different such equivalence classes, so the tensor product of infinitely many copies of $R$ ends up being a huge direct sum of copies of $R$.  (When $R$ is not a field, weirder things can happen with these direct summands--for instance, when $R=\mathbb{Z}$, you also get summands that are isomorphic to all the different subgroups of $\mathbb{Q}$.)
In particular, this can be viewed as a failure of associativity: the empty tensor product is just $R$, and the empty set can be written as a disjoint union of infinitely many copies of the empty set, and we have $$\bigotimes_{i\in\emptyset}M_i\not\cong\bigotimes_{p\in P}\bigotimes_{i\in I_p}M_i$$ where $P$ is infinite and each $I_p$ is empty, since the left side is just $R$ but the right side is an infinite tensor product of copies of $R$.
There are similar failures of associativity using nonempty sets, though they are a bit more complicated to describe.  For instance, suppose we partition $I$ into doubleton sets $I_p$, where each tensor product $\bigotimes_{i\in I_p}M_i$ contains a non-simple tensor $x_p$, i.e. an element that cannot be written just as a tensor product $a\otimes b$ but instead must be written as a sum of multiple such products.  The product $\bigotimes_p\bigotimes_{i\in I_p}M_i$ then has an element $\bigotimes x_p$ that is the tensor product of each of these elements $x_p$.  This element $\bigotimes x_p$ should not correspond to any element of $\bigotimes_{i\in I}M_i$, since if you were to expand it out as a sum of tensors $\bigotimes m_i$ where $m_i\in M_i$, there would be infinitely many terms (each $x_p$ expands out to a sum of more than one term, and then you want to multiply infinitely many such sums together but that would produce an infinite sum).  Turning this idea into a rigorous proof that $\bigotimes x_p$ is not in the image of the canonical map $\bigotimes_{i\in I}M_i\to\bigotimes_p\bigotimes_{i\in I_p}M_i$ in some particular example takes a bit more work and requires a similar analysis of which tensors can be related to each other by a finite sequence of relations as in the first example above.
