# Tensor product of modules and algebras

It seems like tensor product of modules over a ring and tensor product of algebras over a ring are studied somewhat independently, is it possible to see these tensor product as one single phenomenon ? For example a tensor product of non-unitary algebras over a ring : modules been then non-unitary algebras with null multiplication, algebras are non-unitary algebras having a unit (but the embedding of category is not fully faithful, that should be a problem I think). Anyone know how to do that ?

This question is motivated by the following : is it possible de interpret the the tensor product of modules as the coproduct in a richer category (my guess : non-unitary algebras) ? My final motivation is : how can be define in general a tensor product "indexed by" any category ? While the tensor product of a finite number of modules can be seen as a tensor product over a finite set, what is a good notion of tensor product indexed by an infinite set ? Or a general category ?

Categorical definitions of $$\otimes$$ have been given in the context of symmetric monoidal categories. However be aware the obvious model is $$\otimes_A$$ where $$A$$ is commutative. Non-commutative $$A$$ wont quite fit that model nor will multiple $$A'$$s. In particular $$U_1\otimes_{A_1}U_2\otimes_{A_2} U_3$$ isn't even well-defined. Does one mean $$(U_1\otimes_{A_1}U_2)\otimes_{A_2} U_3$$, in which case $$A_2$$ acts on $$U_1\otimes_{A_1} U_2$$, or does it mean $$U_1\otimes_{A_1}(U_2\otimes_{A_2} U_3)$$ where now $$A_1$$ acts on $$(U_2\otimes_{A_2} U_3)$$? Most authors intend a third idea, which is that $$U_2$$ is an $$(A_1,A_2)$$-bimodule and thus $$U_1\otimes_{A_1}U_2$$ becomes an $$A_2$$-module and $$U_2\otimes_{A_2} U_3$$ an $$A_1$$-module; so, some form of associativity works. But one sees the defects in thinking of such a definition as an $$n$$-ary product. So even the concept of a tensor over a set as suggested needs shoring up before further generalizations. This in part is why the axioms of symmetric monoidal categories assume a single tensor product.
It turns out that if you form $$U\otimes_{\Delta} V$$ for any set $$\Delta$$ operating linearly on $$U$$ and $$V$$ (formally there is a function $$\Delta\to \mathrm{End}(U)\times \mathrm{End}(V)^{op}$$), then in fact you can effectively "close" $$\Delta$$ to an associative ring, an adjoint algebra $$A(\Delta)$$. And $$U\otimes_{A(\Delta)} V=U\otimes_{\Delta} V$$ -- equal not just isomorphic. So in this sense for a binary tensor product any creative set of operations will result in the same as just using associative algebras; see Theorem 2.11 in the below reference.
There is a recent generalization of that result for $$n$$-ary tensor products, but the result is that the closure of the operators you tensor over is a Lie algebra, not an associative algebra. In short, tensoring over categories -- whatever that should mean -- probably wont get you what you want. You need a Lie structure, and categories don't have that.