# Another characterization of tensor products of modules

It is known that the tensor product is characterized by its universality in the category of $$A$$-modules.

Does the following proposition hold?

Proposition There exists only one operation $$\otimes$$ that satisfies the following four properties in the category of $$A$$-modules.

Let $$M$$, $$N$$, and $$P$$ be $$A$$-modules. Then

(i) $$M\otimes N\cong N\otimes M$$

(ii) $$(M\otimes N)\otimes P\cong M\otimes (N\otimes P)$$

(iii) $$(M\oplus N)\otimes P\cong (M\otimes P)\oplus (N\otimes P)$$

(iv) $$A\otimes M\cong M$$

• This formulation is a priori strange, as you do not impose any functoriality on these isomorphisms.
– Z. M
Mar 20 at 17:55

As mentioned in the comments there is likely more you should require about your structure, like functoriality and the coherence/compatibility conditions for the properties that you are proposing here.

In other words, you're asking if the tensor product of modules is the unique additive symmetric monoidal category structure on A-Mod with unit given by the A-module A.

In this other answer for a related question Abelian categories that are not monoidal I referenced the work of Hovey, in which he shows that the only closed symmetric additive monoidal category structure on A-Mod with unit isomorphic to A ( for commutative A ), is the usual such structure.

So, if you're willing to add the conditions explained above and put the reasonable hypothesis of the structure being closed, then the answer is yes.

• Being closed is very strong, which implies that the tensor product preserves small colimits separatedly in each variable, while it seems to me that the OP would only assume that it preserves (finite nonempty) direct sums.
– Z. M
Mar 20 at 21:04

Let $$A=$$ a field, $$m = \dim M$$, $$n = \dim N$$. Define $$M \otimes N$$ by $$\dim M \otimes N = mn$$ if $$m$$ is finite or $$n$$ is finite, otherwise $$\kappa mn$$, for some fixed infinite cardinal $$\kappa$$.