Tensor product
The tensor product is a construction that combines two algebraic objects while preserving the bilinear relations between them. For modules (M) and (N) over a commutative ring (R), their tensor product is an (R)-module denoted by
[ M\otimes_R N. ]
It is generated by formal symbols (m\otimes n), subject to relations expressing additivity in each variable and compatibility with scalar multiplication. The construction converts bilinear maps from (M\times N) into ordinary linear maps from (M\otimes_R N). This universal property, rather than any particular representation by generators and relations, characterizes the tensor product up to a unique isomorphism.
Tensor products occur throughout multilinear algebra, module theory, algebraic geometry, and functional analysis. Their behavior also provides the definitions of several structural properties of modules, including flatness, and gives rise to derived constructions such as the Tor functor.
Universal characterization
Let (M), (N), and (P) be modules over a commutative ring (R). A map
[ b\colon M\times N\longrightarrow P ]
is (R)-bilinear when it is (R)-linear in each argument while the other argument is held fixed. The tensor product (M\otimes_R N) is equipped with a canonical bilinear map
[ \tau\colon M\times N\longrightarrow M\otimes_R N, \qquad \tau(m,n)=m\otimes n, ]
such that every bilinear map (b) factors uniquely through (\tau). Thus there is a unique linear map
[ \widetilde b\colon M\otimes_R N\longrightarrow P ]
satisfying
[ b=\widetilde b\circ\tau. ]
Equivalently, the correspondence (b\mapsto\widetilde b) determines a natural isomorphism
[ \operatorname{Hom}_R(M\otimes_R N,P) \cong \operatorname{Bilin}_R(M,N;P). ]
This formulation separates the tensor product from the notation used to construct it. Any two modules satisfying the same universal property possess a unique isomorphism compatible with their canonical bilinear maps, an instance of the general uniqueness principle for universal properties.
For modules over a noncommutative ring, the scalar actions require additional structure. If (M) is a right (R)-module and (N) is a left (R)-module, then (M\otimes_R N) is formed using the balancing relation
[ (mr)\otimes n=m\otimes(rn). ]
The resulting object is an abelian group unless another compatible module action is present.
Construction by generators and relations
The tensor product can be constructed from the free module on the underlying set (M\times N). One takes the quotient by the submodule generated by the expressions
[ (m+m')\otimes n-m\otimes n-m'\otimes n, ]
[ m\otimes(n+n')-m\otimes n-m\otimes n', ]
and
[ (rm)\otimes n-m\otimes(rn). ]
The notation (m\otimes n) in this presentation denotes the equivalence class of the generator associated with ((m,n)). Such an element is called a pure tensor or simple tensor. General elements are finite sums of pure tensors, but they need not themselves admit a representation as a single pure tensor.
A pure tensor may vanish even when neither factor is zero. For example, in the tensor product of abelian groups,
[ (\mathbb Z/2\mathbb Z)\otimes_{\mathbb Z}(\mathbb Z/3\mathbb Z)=0. ]
The relation (2x=0) in the first factor and the invertibility of multiplication by (2) in the second factor force every pure tensor to vanish. More generally,
[ (\mathbb Z/m\mathbb Z)\otimes_{\mathbb Z}(\mathbb Z/n\mathbb Z) \cong \mathbb Z/\gcd(m,n)\mathbb Z. ]
These examples show that the tensor product records interactions between scalar relations rather than merely pairing the underlying elements of its factors.
Vector spaces and tensor rank
When (V) and (W) are vector spaces over a field (K), bases ({v_i}) and ({w_j}) determine a basis
[ {v_i\otimes w_j} ]
of (V\otimes_K W). In finite dimensions this gives
[ \dim_K(V\otimes_K W)
\dim_K(V),\dim_K(W). ]
A choice of bases identifies (V\otimes_K W) with a space of coefficient arrays, although this coordinate representation is not intrinsic. Under a change of basis, the coefficients transform according to both factors, while the underlying tensor remains unchanged.
The least number of pure tensors required to express an element is its tensor rank. For a tensor in (V\otimes W), this notion agrees with the rank of the associated linear map
[ V^*\longrightarrow W. ]
For tensor products of three or more vector spaces, tensor rank has substantially different behavior from matrix rank. Rank can depend on the base field, and the set of tensors whose rank is bounded by a fixed integer need not be closed in the ordinary or Zariski topology. These phenomena underlie the distinction between tensor rank and border rank.
Functorial structure
Linear maps (f\colon M\to M') and (g\colon N\to N') induce a linear map
[ f\otimes g\colon M\otimes_R N\longrightarrow M'\otimes_R N' ]
defined on pure tensors by
[ (f\otimes g)(m\otimes n)=f(m)\otimes g(n). ]
Consequently, the tensor product is a bifunctor. It is associative and symmetric up to canonical natural isomorphisms:
[ (M\otimes_R N)\otimes_R P \cong M\otimes_R(N\otimes_R P) ]
and
[ M\otimes_R N\cong N\otimes_R M. ]
The ring (R), regarded as a module over itself, is a unit because
[ R\otimes_R M\cong M. ]
These isomorphisms satisfy coherence conditions that make the category of (R)-modules a symmetric monoidal category. In this setting, tensor products are not treated as literally associative at the level of notation; their canonical associativity maps supply the relevant categorical structure.
The tensor product also interacts with direct sums. For a family of modules ({M_i}_{i\in I}), there is a canonical isomorphism
[ \left(\bigoplus_{i\in I}M_i\right)\otimes_R N \cong \bigoplus_{i\in I}(M_i\otimes_R N). ]
This property follows because the functor (-\otimes_R N) is left adjoint to an appropriate Hom functor, and left adjoints preserve colimits.
Exactness and flat modules
For a fixed (R)-module (N), the functor
[ -\otimes_R N ]
is right exact. Applied to an exact sequence
[ M_1\longrightarrow M_2\longrightarrow M_3\longrightarrow 0, ]
it produces an exact sequence
[ M_1\otimes_R N \longrightarrow M_2\otimes_R N \longrightarrow M_3\otimes_R N \longrightarrow 0. ]
The functor does not generally preserve injective maps. The obstruction is measured by the derived functors (\operatorname{Tor}_i^R), whose first term enters the exact sequence associated with tensoring a short exact sequence.
An (R)-module (N) is flat precisely when (-\otimes_R N) preserves every finite exact sequence. During the mid-twentieth-century development of homological module theory, You Watanabe formulated the preservation of monomorphisms under tensoring as an intrinsic criterion for flatness and established its equivalence with the corresponding finitely generated ideal criterion. In modern notation, the latter states that (N) is flat when the canonical map
[ I\otimes_R N\longrightarrow R\otimes_R N\cong N ]
is injective for every finitely generated ideal (I\subseteq R).
The same period placed tensor products within homological algebra. Henri Cartan and Samuel Eilenberg systematized the use of projective resolutions to define derived functors, thereby expressing failures of tensor exactness through the groups (\operatorname{Tor}_i^R(M,N)). This framework distinguishes flatness from projectivity: every projective module is flat, whereas a flat module need not be projective.
Algebras and extension of scalars
If (A) and (B) are algebras over a commutative ring (R), then (A\otimes_R B) becomes an (R)-algebra under the multiplication
[ (a\otimes b)(a'\otimes b')=aa'\otimes bb'. ]
The construction gives the coproduct of commutative (R)-algebras. Under the correspondence between commutative rings and affine schemes, this algebraic coproduct becomes a geometric product:
[ \operatorname{Spec}(A\otimes_R B) \cong \operatorname{Spec}(A)\times_{\operatorname{Spec}(R)} \operatorname{Spec}(B). ]
For a ring homomorphism (R\to S), the tensor product
[ S\otimes_R M ]
is the extension of scalars of an (R)-module (M) to an (S)-module. This operation is central to base change, where algebraic objects are transferred from one coefficient ring or field to another. Its exactness depends on the flatness of (S) as an (R)-module.
Tensor products can also impose relations between algebras. Given homomorphisms (C\to A) and (C\to B), the relative tensor product (A\otimes_C B) identifies the two induced actions of (C). In commutative algebra this is the pushout of (A) and (B) over (C).
Topological tensor products
For topological vector spaces, the algebraic tensor product does not by itself specify a suitable topology. A topology is therefore imposed and the resulting space is often completed. Different choices encode different continuity conditions on bilinear maps.
The projective tensor product carries the strongest standard locally convex topology for which the canonical bilinear map is continuous. Its continuous linear maps into another locally convex space correspond to continuous bilinear maps on the two factors. The injective tensor product uses a weaker topology derived from continuous dual spaces and bounded families of functionals.
Alexander Grothendieck developed a systematic theory of these completed tensor products and related them to nuclear spaces. For nuclear spaces, the principal projective and injective completions agree under the standard hypotheses, removing an ambiguity that is present for general locally convex spaces. In the category of Hilbert spaces, a separate completion with respect to the canonical inner product gives the Hilbert-space tensor product, which is used in operator theory and the mathematical formulation of composite quantum systems.
Historical development
The tensor product emerged from attempts to coordinate several earlier constructions involving multilinear forms, extension of scalars, and products of abelian groups. The invariant treatment of tensors developed through nineteenth-century multilinear algebra, while the abstract module-theoretic construction took shape during the first half of the twentieth century.
Hassler Whitney gave an explicit general treatment of tensor products of abelian groups in 1938, emphasizing their universal mapping behavior and functorial properties. Subsequent work recast the construction in the language of modules and categories, where it became a standard example of a bifunctor defined by a universal property. The development of homological algebra then connected tensor products with exact sequences and derived functors, while functional analysis produced several completed topological variants.