tensor product
The tensor product is the official home where bilinear behaviour becomes ordinary linear behaviour. Faced with a bilinear map B: M × N -> P that resists linear-algebra methods, we build a single new module M ⊗ N together with one universal bilinear map M × N -> M ⊗ N, (m, n) ↦ m ⊗ n, so cleverly that B is recovered from a unique linear map M ⊗ N -> P. In effect we trade two coupled slots for one space in which they have been multiplied together.
Construct M ⊗_R N (over a commutative ring R) as follows: take the free R-module on all symbols (m, n), then quotient by the relations forcing bilinearity — (m + m', n) − (m, n) − (m', n), (m, n + n') − (m, n) − (m, n'), and (r·m, n) − r·(m, n), (m, r·n) − r·(m, n). The image of (m, n) is written m ⊗ n. Elements are finite sums of such pure tensors; not every element is itself pure.
Tensor products are functorial and satisfy clean rules: M ⊗ N ≅ N ⊗ M, R ⊗ M ≅ M, distributivity over direct sums (M ⊕ M') ⊗ N ≅ (M ⊗ N) ⊕ (M' ⊗ N), and dimension multiplies for vector spaces, dim(V ⊗ W) = dim V · dim W. One honest subtlety: tensoring is right exact but not generally left exact — its failure of exactness is measured by the functors Tor, and modules for which tensoring stays exact are exactly the flat modules.
Z/2Z ⊗_Z Z/3Z = 0, because in m ⊗ n one can write m ⊗ n = 3·(m ⊗ n) − 2·(m ⊗ n) = m ⊗ 3n − 2m ⊗ n = 0 − 0 = 0 (using 3n = 0 and 2m = 0). More generally Z/mZ ⊗ Z/nZ ≅ Z/gcd(m, n)Z.
Tensoring coprime cyclic groups annihilates them — a vivid sign that ⊗ is not just a product.
Over a noncommutative ring R one tensors a right R-module with a left R-module, M ⊗_R N, and the result is only an abelian group unless extra structure (bimodules) is present. The clean commutative theory above is the case most often met first.