Tensor Products

Let R be a commutative ring and M and N be R-modules. (We always work with rings having a multiplicative identity and modules are assumed to be unital: 1 ·m = m for all m ∈M .) The direct sum M ⊕N is an addition operation on modules. We introduce here a product operation M ⊗RN , called the tensor product. We will start off by describing what a tensor product of modules is supposed to look like. Rigorous definitions are in Section 3. Tensor products first arose for vector spaces, and this is the only setting where tensor products occur in physics and engineering, so we’ll describe the tensor product of vector spaces first. Let V and W be vector spaces over a field K, and choose bases {ei} for V and {fj} for W . The tensor product V ⊗K W is defined to be the K-vector space with a basis of formal symbols ei⊗fj (we declare these new symbols to be linearly independent by definition). Thus V ⊗K W is the formal sums ∑ i,j cijei ⊗ fj with cij ∈ K, which are called tensors. Moreover, for any v ∈ V and w ∈W we define v⊗w to be the element of V ⊗KW obtained by writing v and w in terms of the original bases of V and W and then expanding out v ⊗ w as if ⊗ were a noncommutative product (allowing any scalars to be pulled out). For example, let V = W = R2 = Re1 + Re2, where {e1, e2} is the standard basis. (We use the same basis for both copies of R2.) Then R2 ⊗R R2 is a 4-dimensional space with basis e1 ⊗ e1, e1 ⊗ e2, e2 ⊗ e1, and e2 ⊗ e2. If v = e1 − e2 and w = e1 + 2e2, then (1.1) v ⊗ w = (e1 − e2)⊗ (e1 + 2e2) := e1 ⊗ e1 + 2e1 ⊗ e2 − e2 ⊗ e1 − 2e2 ⊗ e2. Does v ⊗ w depend on the choice of a basis of R2? As a test, pick another basis, say e1 = e1 + e2 and e ′ 2 = 2e1 − e2. Then v and w can be written as v = −13e ′ 1 + 2 3e ′ 2 and w = 53e ′ 1 − 13e ′ 2. By a formal calculation,