We can define a tensor product of two vector spaces. But vector spaces are themselves modules and we can also define a tensor product of two modules. My question is the following: are modules the "most generic" algebraic structures for which we can define a tensor product ?
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We can define tensor products for any commutative algebraic theory. This includes $A$-modules for any commutative ring $A$. – Zhen Lin Mar 02 '14 at 15:56
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3Arguably, for any category that can be given the structure of a monoidal category (Wikipedia), its objects "have a tensor product" (namely, the monoidal product). And any category with finite products can be given the structure of a monoidal category, with the Cartesian product playing the role of the "tensor product". Is that satisfactory? – Zev Chonoles Mar 02 '14 at 15:59
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@ZevChonoles I am not very fluent in category theory, but is it correct to say that we can always define a tensor product of two monoids? – Vincent Mar 02 '14 at 16:33
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Well, yes, but that is not really what I am saying in my comment. What my comment is saying is that for any sort of mathematical object where you can form a product $\times$ (e.g., monoids, topological spaces, sets, etc.), it meets all of the abstract properties you would expect from a "tensor product". – Zev Chonoles Mar 02 '14 at 16:35
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@ZevChonoles And in set theory (not category theory), what is the most generic mathematical object where we can form a product ? – Vincent Mar 02 '14 at 16:37
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2@Vincent: That question makes no sense. – Zev Chonoles Mar 02 '14 at 16:41
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@ZevChonoles I expect tensor products of algebraic structures to have a universal property with respect to bihomomorphisms. This is quite a strong condition. – Zhen Lin Mar 03 '14 at 00:14
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The question is answered in the body of Tensor product of monoids and arbitrary algebraic structures – Martin Brandenburg Mar 03 '14 at 11:52