Engineering · Mathematics · Advanced Algebra Handbook
Tensor Products, Flatness and Multilinear Methods
Modules generalise vector spaces by allowing coefficients from a ring; category language then organises objects, morphisms and universal constructions. The central discipline is to distinguish element calculations from map-level or universal properties. This page consolidates the relevant material from the supplied algebra source into a stand-alone handbook chapter.
Executive summary
This chapter develops tensor products, flatness and multilinear methods as part of a connected advanced-algebra learning sequence. The emphasis is on definitions, hypotheses, structural results and repeatable methods rather than historical narrative.
The source material is theorem-rich. Accordingly, the handbook presentation separates vocabulary from results and then adds a verification workflow so that each statement can be applied safely. Mathematical examples in the source are treated as examples, not as universal rules.
Problem-solving workflow
Identify the coefficient ring and whether modules are left, right or bimodules.
State the maps and verify linearity before using kernels, images or exactness.
Use exact sequences to record how subobjects and quotients fit together.
When a construction is defined universally, verify both existence and uniqueness of the mediating map.
For projective or injective arguments, convert lifting or extension properties into split exact sequences where possible.
For limits or colimits, track the direction of every structure map.
Core definitions
Definition
Let R be a ring, let AR be a right R-module, let R B be a left R-module, and let G be an (additive) abelian group. A function f : A × B →G is called R-biadditive if, for all a, a′ ∈A, b, b′ ∈B, and r ∈R, we have f (a + a′, b) = f (a, b) + f (a′, b); f (a, b + b′) = f (a, b) + f (a, b′); f (ar, b) = f (a,rb). An R-biadditive function is also called a pairing. If R is commutative and A, B, and M are R-modules, then a function f : A × B →M is called R-bilinear if f is R-biadditive and also f (ar, b) = f (a,rb) = r f (a, b).
Definition
Given a ring R and modules AR and R B, then their tensor product is an abelian group A ⊗R B and an R-biadditive function h : A × B →A ⊗R B such that, for every abelian group G and every R-biadditive f : A × B →G, there exists a unique Z-homomorphism Δf : A ⊗R B →G making the following diagram commute. Quite often, we denote A ⊗R B by A ⊗B when R = Z.
Definition
Let R and S be rings and let M be an abelian group. Then M is an (R, S)- bimodule, denoted by R MS, if M is a left R-module and a right S-module, and the two scalar multiplications are related by an associative law: r(ms) = (rm)s for all r ∈R, m ∈M, and s ∈S. If M is an (R, S)-bimodule, it is permissible to write rms with no parentheses, for the definition of bimodule says that the two possible associations agree.
Definition
If k is a commutative ring, then a k-bilinear product is a k-module X and a k-bilinear function h : A × B →X such that, for every k-module M and every k-bilinear function g : A × B →M, there exists a unique k-homomorphism g: X →M making the following diagram commute. A × B h g → X g M The next result shows that k-bilinear products exist, but that they are nothing new.
Definition
If R is a ring, then a right R-module A is flat8 if, whenever 0 →B′ i→B p→B′′ →0 8 This term arose as the translation into algebra of a geometric property of varieties. Tensor Products is an exact sequence of left R-modules, then 0 →A ⊗R B′ 1A⊗i −→A ⊗R B 1A⊗p −→A ⊗R B′′ →0 is an exact sequence of abelian groups. Flatness of a left R-module is defined similarly. In other words, A is flat if and only if A⊗R is an exact functor.
Definition
If B is a right R-module, define its character module B∗as the left R-module B∗= HomZ(B, Q/Z). Recall that B∗is a left R-module if one defines r f , for r ∈R and f : B →Q/Z, by r f : b ↦f (br). The next lemma improves Proposition 7.48: If i : A′ →A and p: A →A′′ are maps and, for every module B, 0 →Hom(A′′, B) p∗ −→Hom(A, B) i∗ −→Hom(A′, B) is an exact sequence, then so is A′ i −→A p −→A′′ →0.
Definition
A module P is small if the covariant Hom functor Hom(P, ) preserves (possibly infinite) direct sums. For example, Proposition 8.85 shows that every ring R is a small R-module. To say that P is small means more than that there is some isomorphism Hom ( P, i∈I Bi ) ∼= i∈I Hom(P, Bi); it also means that Hom(P, ) preserves the coproduct diagram; if λi : Bi →B are the injections, where B = i∈I Bi, then the induced maps (λi)∗: Hom(P, Bi) →Hom(P, B) are the injections of i∈I Hom(P, Bi).
Definition
A right R-module P is a generator of ModR if every right R-module M is a quotient of some direct sum of copies of P. It is clear that R is a generator of ModR, as is any free right R-module. However, a projective right R-module may not be a generator. For example, if R = I6, then R = P ⊕Q, where P = {[0], [2], [4]} ∼= I3, and the projective module P is not a generator (for Q ∼= I2 is not a quotient of a direct sum of copies of P).
Principal results and structural facts
Key result
Let f : AR →A′ R and g : R B →R B′ be maps of right R-modules and left R-modules, respectively. Then there is a unique Z-homomorphism, denoted by f ⊗g : A ⊗R B →A′ ⊗R B′, with f ⊗g : a ⊗b ↦f (a) ⊗g(b).
Key result
If f : M →M′ and g : N →N ′ are, respectively, isomorphisms of right and left R-modules, then f ⊗g : M ⊗R N →M′ ⊗R N ′ is an isomorphism of abelian groups. Tensor Products
Key result
now gives r(a ⊗b) = (ra) ⊗b = (ar) ⊗b = a ⊗rb. (iii) This statement merely sees the last equation a ⊗rb = r(a ⊗b) from a different viewpoint: (1A ⊗µr)(a ⊗b) = a ⊗rb = r(a ⊗b). • We have defined R-biadditive functions for arbitrary, possibly noncommutative, rings R, whereas we have defined R-bilinear functions only for commutative rings. Tensor product was defined as the solution of a certain universal mapping problem involving R-biadditive functions; we now consider the analogous problem for R-bilinear functions when R is commutative. Here is a provisional definition, soon to be seen unnecessary.
Key result
If k is a commutative ring and M and N are k-modules, then there is a k-isomorphism τ : M ⊗k N →N ⊗k M with τ : m ⊗n ↦n ⊗m.
Key result
For every left R-module M, there is an R-isomorphism θM : R ⊗R M →M with θM : r ⊗m ↦rm. Indeed, θ = {θM} is a natural equivalence between R⊗R and the identity functor on RMod.
Key result
Given a commutative diagram with exact rows in which the vertical maps f and g are isomorphisms, A′ i f A p g A′′ h B′ j B q B′′ 0, there exists a unique isomorphism h : A′′ →B′′ making the augmented diagram commute.
Key result
If D is a nonzero divisible abelian group with every element of finite order (e.g., D = Q/Z), then there is no multiplication D × D →D making D a ring.
Key result
); thus, S is generated by all elements in F of the form (m, a + a′) −(m, a) −(m, a′); (m + m′, a) −(m, a) −(m′, a); (mr, a) −(m,ra). Let M′ be the submodule of M generated by x1, . . . , xn together with the (finite number of) first “coordinates” in M exhibiting k(x j, iy j) as a linear combination of relators just displayed. Of course, M′ is a finitely generated submodule of M. The element u′ = x j ⊗y j ∈M′⊗R A (which is the version of u lying in this new tensor product M′⊗R A) lies in ker 1M′ ⊗i, for we have taken care that all the relations making (1M ⊗i)(u) = 0 are still present. But M′ is a finitely generated submodule of M, so that it is flat, by hypothesis, and so (1M′⊗i)(u) = 0 implies u′ = 0 in M′⊗R A. Finally, if ℓ: M′ →M is the inclusion, then (ℓ⊗1A)(u′) = u, and so u = 0. Therefore, 1M ⊗i is injective and M is flat. Here are some examples of flat modules.
Key result
Let A be a right R-module, and let B′ i→B p→B′′ →0 be an exact sequence of left R-modules. Then A ⊗R B′ 1A⊗i −→A ⊗R B 1A⊗p −→A ⊗R B′′ →0 is an exact sequence of abelian groups.
Key result
(i) If R is a domain with Q = Frac(R), then Q is a flat R-module. (ii) If every finitely generated submodule of a right R-module M is flat, then M is flat.
Key result
A sequence of right R-modules 0 →A α −→B β −→C →0 is exact if and only if the sequence of character modules 0 →C∗ β∗ −→B∗ α∗ −→A∗→0 is exact.
Key result
A right R-module B is flat if and only if, for every finitely generated left ideal I, the sequence 0 →B ⊗R I →B ⊗R R is exact.
Key result
If R is right ascending-chain-finite, then a finitely generated right R-module B is flat if and only if it is projective.
Key result
Every group G of order pmqn, where p and q are primes, is a solvable group. Notice that finite-group solvability theorem cannot be improved to groups having orders with only three distinct prime factors, for A5 is a simple group of order 60 = 22 · 3 · 5. Using representations, we will prove the following theorem. Theorem. If G is a nonabelian finite simple group, then {1} is the only conjugacy class whose size is a prime power.
Source-grounded examples
Worked source example
(i) If R is a ring, then its multiplication µ: R × R →R is R-biadditive; the first two axioms are the right and left distributive laws, while the third axiom is associativity: µ(ar, b) = (ar)b = a(rb) = µ(a,rb). If R is a commutative ring, then µ is R-bilinear, for (ar)b = a(rb) = r(ab). (ii) If R M is a left R-module, then its scalar multiplication σ : R×M →M is R-biadditive; if R is a commutative ring, then σ is R-bilinear. (iii) If MR and NR are right R-modules, then HomR(M, N) is a left R-module if, for f ∈HomR(M, N) and r ∈R, we define r f : M →N by r f : m ↦f (mr). The dual space V ∗of a vector space V over a field k gives a special case of this construction: Evaluation V × V ∗→k is R-bilinear. ◀ Tensor products convert biadditive functions into linear ones.
Worked source example
(i) Any finite direct sum of small modules is small, and any direct summand of a small module is small. (ii) Since a ring R is a small R-module, it follows from (i) that every finitely generated free R-module is small and that every finitely generated projective R-module is small. ◀
How to reason with these results
Most advanced-algebra problems become manageable when the representation is separated from the invariant structure. Begin with the definition, then decide whether the problem is asking for an elementwise calculation, a statement about a morphism, or a classification up to isomorphism. That choice determines the correct proof language.
When a theorem gives a structural conclusion, do not jump directly to the conclusion. Write the hypotheses next to the object you are studying and check them one by one. If a hypothesis fails, either strengthen the object, pass to a quotient or localisation where the theorem applies, or use a more elementary argument.
For computational work, record each transformation together with the equivalence relation it preserves. In algebra, row operations, similarity, quotienting, localisation and isomorphism preserve different kinds of information. A calculation is useful only when the preserved structure matches the question.
Common failure modes
| Failure mode | Control |
|---|---|
| Treating a module as a vector space when the coefficient ring is not a field. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Reversing arrows in contravariant constructions. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Calling a sequence exact without checking equality of image and kernel at each object. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Using a universal construction without proving uniqueness. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Confusing direct products with direct sums in infinite families. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
Verification checklist
- The ambient set, ring, field, group, module or category has been stated.
- Every operation and map used is well-defined in that setting.
- The hypotheses of each structural result have been checked before use.
- Representatives, coordinates or generators have not been confused with the underlying object.
- Existence and uniqueness have been separated where both matter.
- The final result has been checked against the original defining relation or universal property.
Quick questions
What should I identify first in a problem about tensor products, flatness and multilinear methods?
Start with the ambient algebraic structure, its operation or maps, and the exact hypotheses. Most incorrect solutions begin by using a familiar rule that is not valid in the stated structure.
How should definitions be used in proofs?
Expand the definition at the point where it becomes useful. Definitions are not background prose; they are the conditions that determine what must be proved and which implications are available.
When is a structural theorem safer than direct calculation?
Use a structural theorem when its hypotheses are satisfied and the calculation would otherwise depend on arbitrary coordinates, representatives or generators. The theorem usually identifies an invariant that survives those choices.
How can a final answer be checked?
Substitute the result back into the defining relation, verify any required closure or map property, and check edge cases such as zero, the identity, the empty object or degenerate quotients where relevant.