Tensor Products of Modules
Handbook guide to tensor products of modules with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Tensor Product of Modules Over a Commutative Ring
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
General Tensor Products
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Motivation In many areas of algebra and its applications, it is useful to multiply, in
Motivation In many areas of algebra and its applications, it is useful to multiply, in a sensible way, an element x of an R-module M by an element y of an R-module N. In group representation theory, M and N are free modules, in fact finite-dimensional vector spaces, with bases {xi} and {yj}. Thus if we specify that multiplication is linear in each variable, then we need only specify products of xi and yj. We require that the these products, to be denoted by xi ⊗yj, form a basis for a new R-module T.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Definition
Let M and N be arbitrary R-modules, and let F be a free R-module with basis M ×N. Let G be the submodule of F generated by the “relations” (x + x′, y) −(x, y) −(x′, y); (x, y + y′) −(x, y) −(x, y′); (rx, y) −r(x, y); (x, ry) −r(x, y) where x, x′ ∈M, y, y′ ∈N, r ∈R. Define the tensor product of M and N (over R) as T = M ⊗R N = F/G and denote the element (x, y) + G of T by x ⊗y. Thus the general element of T is a finite sum of the form t = i xi ⊗yi (1) with xi ∈M and yi ∈N.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition Regarding R as a module over itself, R ⊗R M ∼= M.
Regarding R as a module over itself, R ⊗R M ∼= M.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Corollary
Let Rm be the direct sum of m copies of R, and M m the direct sum of m copies of M. Then Rm ⊗M ∼= M m.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Proposition
Rm ⊗Rn ∼= Rmn. Moreover, if {x1, . . . , xm} is a basis for Rm and {y1, . . . , yn} is a basis for Rn, then {xi ⊗yj, i = 1, . . . , m, j = 1, . . . , n} is a basis for Rmn.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Definition
Let M be a right R-module and N a left R-module. (We often abbreviate this as MR and RN.) Let f : M ×N →P, where P is an abelian group. The map f is biadditive if it is additive in each variable, that is, f(x+x′, y) = f(x, y)+f(x′, y) and f(x, y + y′) = f(x, y) + f(x, y′) for all x, x′ ∈M, y, y′ ∈N. The map f is R-balanced if f(xr, y) = f(x, ry) for all x ∈M, y ∈N, r ∈R.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Construction of the General Tensor Product If MR and RN, let F be the free
of the General Tensor Product If MR and RN, let F be the free abelian group with basis M × N. Let G be the subgroup of R generated by the relations (x + x′, y) −(x, y) −(x′, y); (x, y + y′) −(x, y) −(x, y′); (xr, y) −(x, ry) where x, x′ ∈M, y, y′ ∈N, r ∈R. Define the tensor product of M and N over R as T = M ⊗R N = F/G and denote the element (x, y) + G of T by x ⊗y. Thus the general element of T is a finite sum of the form t = i xi ⊗yi.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Bimodules Let R and S be arbitrary rings. We say that M is an S −R bimodule if
Bimodules Let R and S be arbitrary rings. Call M is an S −R bimodule if M is both a left S-module and a right R-module, and in addition a compatibility condition is satisfied: (sx)r = s(xr) for all s ∈S, r ∈R. We often abbreviate this as SMR. If f : R →S is a ring homomorphism, then S is a left S-module, and also a right R-module by restriction of scalars, as in (8.7.1).
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Proposition
If SMR and RNT , then M ⊗R N is an S −T bimodule.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Corollary
If SMR and RN, then M ⊗R N is a left S-module. If MR and RNT , then M ⊗R N is a right T-module.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Extensions As in Section 8.7, we can define the tensor product of any finite number
Extensions As in Section 8.7, one can define the tensor product of any finite number of modules using multiadditive maps (additive in each variable) that are balanced. For example, suppose that MR, RNS and SP. If f : M × N × P →G, where G is an abelian group, the condition of balance is f(xr, y, z) = f(x, ry, z) and f(x, ys, z) = f(x, y, sz) for all x ∈M, y ∈N, z ∈P, r ∈R, s ∈S.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Tensor Product of Algebras
Tensor Product of Algebras If A and B are algebras over the commutative ring R, then the tensor product A ⊗R B becomes an R-algebra if define multiplication appropriately. Consider the map of A × B × A × B into A ⊗R B given by (a, b, a′, b′) →aa′ ⊗bb′, a, a′ ∈A, b, b′ ∈B. The map is 4-linear, so it factors through the tensor product to give an R-module homomorphism g : A ⊗B ⊗A ⊗B →A ⊗B such that g(a ⊗b ⊗a′ ⊗b′) = aa′ ⊗bb′. Now let h : (A ⊗B) × (A ⊗B) →A ⊗B ⊗A ⊗B be the bilinear map given by h(u, v) = u ⊗v.
Proof / verification strategy: Assume a non-trivial factorisation or divisibility relation and use degree, content, ideal or prime-divisibility constraints to force one factor to be a unit or to obtain a contradiction.
Quick-reference relationships
Problem-solving workflow
Fix the coefficient ring and variance
State whether modules are left/right modules and whether a functor is covariant or contravariant.
Write the maps, not just the objects
Kernels, images, exactness and universal properties depend on the actual homomorphisms.
Use the appropriate universal property
Direct sums, products, tensor products, projectives, injectives and limits are best handled by their mapping property.
Check exactness at each position
Verify image equals kernel rather than relying on the appearance of a diagram.
Choose a resolution only when needed
Derived constructions should be tied to projective or injective resolutions and independence from the chosen resolution.
Test naturality and compatibility
For induced maps, ensure compositions and commutative squares behave as required.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test order, kernel, ideal, polynomial, basis, module, homomorphism, tensor. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Forgetting the coefficient ring when comparing modules.
- Assuming tensor product preserves every exact sequence.
- Confusing direct sum with direct product for infinite families.
- Reading exactness from a diagram without checking image equals kernel.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 8.7 | Tensor Product of Modules Over a Commutative Ring | 170–173 |
| 8.8 | General Tensor Products | 174–177 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.