Tensoring

Tensoring is the operation of forming the tensor product of algebraic objects such as vector spaces or modules over a ring. Given a commutative ring R and left R-modules N and M, the tensor product N ⊗_R M is another R-module that encodes bilinear information about N and M. The construction is functorial in each argument, and for a fixed module M it defines a functor - ⊗_R M from the category of R-modules to itself.

The tensor product is characterized by a universal property: there is a natural correspondence between bilinear maps N × M → L and linear maps N ⊗_R M → L. This universal property determines the tensor product up to a unique isomorphism and explains why tensoring converts bilinear information into linear maps.

Key properties include associativity and the unit object. There are isomorphisms (N ⊗_R M) ⊗_R P ≅ N ⊗_R (M ⊗_R P), and R ⊗_R M ≅ M ≅ M ⊗_R R. Over a field F, if V and W are finite-dimensional, then the dimension of their tensor product satisfies dim_F(V ⊗_F W) = dim_F V · dim_F W.

Exactness and flatness are important aspects of tensoring. Tensoring with a fixed module M is right exact: for any sequence A → B → C → 0, A ⊗_R M → B ⊗_R M → C ⊗_R M → 0 is exact. It need not preserve kernels in general; tensoring is exact for all sequences precisely when M is flat. Over a field, every module is flat, so tensoring is exact in that setting.

Applications of tensoring appear across mathematics: base change in linear algebra, constructions in algebraic topology, and the definition of tensor bundles in differential geometry. Example: V ⊗_F F^n ≅ V^n, illustrating how tensoring scales a space by a fixed finite-dimensional factor.