Tensors in math and physics

Question

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?

Answer 1

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 $\mathbb{R})$.