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I know how tensor product f two modules is defined in communtative algebra. But there is also a concept of tensors used in physics. Are these two concepts related? If yes, can someone explain me tensors as used in physics in terms of tensor product of two module?

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Yes, they are just tensor products of "physically meaningful" modules. – Willie Wong Feb 13 '13 at 16:49

The tensors in physics are usually (I think) elements of the tensor product of two or more vector spaces. Vector spaces are in particular modules over fields, and the tensor product of vector spaces agrees with the more general tensor product of modules.

I think physicists probably also use tensor fields on manifolds, which are slightly more general, in that they give you a tensor in a tensor power of the tangent space to the manifold at each point, but pointwise this is the same construction as above; each tangent space is a module over the base field (usually $\mathbb{R})$.

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Actually, quite often smooth tensor fields on manifolds are constructed as follows, making full use of the modules (instead of vector space) point of view: the set of smooth vector fields on a manifold (section of the tangent bundle) form a module over the ring of smooth functions over the same manifold. Tensor fields are elements of the tensor product of the above module with either itself or with its dual. – Willie Wong Feb 13 '13 at 16:52
Ah, that's great! I hoped that would be true, but didn't know off the top of my head. – Matthew Pressland Feb 13 '13 at 16:54
Using more general modules instead of vector spaces leads exactly to the definition of spinors and spinor fields, in physics. – geodude Jul 25 '13 at 13:12

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