5.15.2 Tensor Product Homogeneity Relation

The Tensor Product Homogeneity Relation ensures degree preservation in tensor algebra by maintaining homogeneity across algebraic structures.

Tensor Product Homogeneity Relation is the generator of the tensor product's defining equivalence relation that governs how scalar multiples interact with the canonical map, guaranteeing that a scalar can be factored out of either argument of a simple tensor and moved freely across the tensor symbol. Together with the additivity relation, it forms the complete set of conditions that convert the free module on pairs (v, w) into the tensor product V ⊗ W, and it is specifically responsible for the scalar, rather than additive, aspect of bilinearity.

The Two Homogeneity Identities

Left Homogeneity

For v ∈ V, w ∈ W, and a scalar c ∈ F, left homogeneity states:

(c v) \otimes w = c (v \otimes w)

Right Homogeneity

Symmetrically, right homogeneity states:

v \otimes (c w) = c (v \otimes w)

Combined Scalar Migration

Together these two identities give the single combined rule most often used in computation:

(c v) \otimes w = v \otimes (c w) = c (v \otimes w)

Realizing Homogeneity in the Quotient Construction

Generators in the Relation Submodule

Within the free module F(V × W), homogeneity is enforced by including in the relation submodule R all elements of the form:

(c u, w) - c (u, w)

(u, c w) - c (u, w)

Passing to the Quotient

Setting these generators to zero forces the equivalence classes of (cu, w) and (u, cw) to both equal c times the equivalence class of (u, w) in the quotient F(V × W)/R, which is exactly the homogeneity relation restated in tensor notation.

Why Homogeneity Must Hold in Both Arguments

Preventing an Asymmetric Structure

If homogeneity were imposed only in the left argument and not the right, the resulting quotient would still allow v ⊗ (cw) and c(v ⊗ w) to be distinct elements, which would break the intended symmetry of the bilinear pairing and prevent many standard identities, such as v ⊗ (cw) = (cv) ⊗ w, from holding.

Consistency with the Field of Scalars

Because V and W are vector spaces over the same field F, homogeneity in both arguments ensures that scalar multiplication on V ⊗ W is unambiguous: multiplying a simple tensor by a scalar gives the same result whether the scalar is absorbed into the left factor, the right factor, or applied directly to the tensor as a whole.

Consequences of Homogeneity

Non-Uniqueness of Factorization into Vectors

A direct consequence of homogeneity is that the pair of vectors producing a given simple tensor is never unique when the field has more than one nonzero element: for any nonzero scalar c, (cv) ⊗ (c⁻¹w) = v ⊗ w, since:

(c v) \otimes (c^{- 1} w) = c c^{- 1} (v \otimes w) = v \otimes w

Compatibility with the Zero Scalar

Setting c = 0 in either homogeneity identity recovers 0 ⊗ w = 0 and v ⊗ 0 = 0, reproducing from the scalar side the same vanishing results that additivity also implies.

Visualization

Role Within the Broader Relation Structure

One Half of the Complete Bilinear Relation Set

The homogeneity relation supplies two of the four total generating families of the relation submodule R, the other two being the left and right additivity relations. Together, additivity and homogeneity exhaust the full content of bilinearity, and no further relations are needed to define V ⊗ W.

Extension to Modules over Noncommutative Rings

When V and W are modules over a noncommutative ring rather than vector spaces over a field, the homogeneity relation must be adapted to track left versus right module actions carefully, since scalar migration across the tensor symbol is only valid when the ring elements involved commute appropriately with the module structures on each side, a subtlety that does not arise in the commutative, field-scalar case described here.