Engineering · Mathematics · Advanced Algebra Handbook
Modules, Submodules and Exact Sequences
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 modules, submodules and exact sequences 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 commutative ring. An R-module is an (additive) abelian group M equipped with a scalar multiplication R × M →M, denoted by (r, m) ↦rm, such that the following axioms hold for all m, m′ ∈M and all r,r′, 1 ∈R: (i) r(m + m′) = rm + rm′; (ii) (r + r′)m = rm + r′m; (iii) (rr′)m = r(r′m); (iv) 1m = m. Remark. This definition also makes sense for noncommutative rings R, in which case M is called a left R-module. ◀
Definition
If M and N are R-modules, then HomR(M, N) = {all R-homomorphisms M →N}. If f, g ∈HomR(M, N), then define f + g : M →N by f + g : m ↦f (m) + g(m).
Definition
If f : M →N is an R-map between R-modules, then kernel f = ker f = {m ∈M : f (m) = 0} and image f = im f = {n ∈N : there exists m ∈M with n = f (m)}. It is routine to check that ker f is a submodule of M and that im f is a submodule of N. Suppose that M = ⟨X⟩; that is, M is generated by a subset X. Suppose further that N is a module and that f, g : M →N are R-homomorphisms. If f and g agree on X [that is, if f (x) = g(x) for all x ∈X], then f = g. The reason is that f −g : M →N, defined by f −g : m ↦f (m) −g(m), is an R-homomorphism with X ⊆ker( f −g). Therefore, M = ⟨X⟩⊆ker( f −g), and so f −g is identically zero; that is, f = g.
Definition
A module M is simple (or irreducible) if M ̸= {0} and M has no proper nonzero submodules; that is, the only submodules of M are {0} and M.
Definition
If S and T are submodules of a module M, then M is their internal direct sum if M ∼= S ⊔T with i : S →M and j : T →M the inclusions. We denote an internal direct sum by M = S ⊕T. Later, we shall write as the mathematical world writes: The same notation S ⊕T is used for either version of direct sum. Here is a restatement of Proposition 7.15 for internal direct sums.
Definition
If S is a submodule of an R-module M, then S is a retract of M if there exists an R-homomorphism ρ : M →S, called a retraction, with ρ(s) = s for all s ∈S.
Definition
A short exact sequence is an exact sequence of the form 0 →A f→B g→C →0. We also call this short exact sequence an extension of A by C. Some authors call this an extension of C by A; some authors say that the middle module B is an extension.
Definition
A category C consists of three ingredients: a class obj(C) of objects, a set of morphisms Hom(A, B) for every ordered pair (A, B) of objects, and composition Hom(A, B) × Hom(B, C) →Hom(A, C), denoted by ( f, g) ↦g f, for every ordered triple A, B, C of objects. [We often write f : A →B or A f→B to denote f ∈Hom(A, B).] These ingredients are subject to the following axioms: (i) the Hom sets are pairwise disjoint;5 that is, each morphism has a unique domain and a unique target; 5One can force pairwise disjointness by labeling morphisms f ∈Hom(A, B) by A f B.
Principal results and structural facts
Key result
Let V be a vector space over a field k, and let T, S : V →V be linear transformations. Then the k[x]-modules V T and V S in Example 7.1(v) are k[x]- isomorphic if and only if there is a vector space isomorphism ϕ : V →V with S = ϕT ϕ−1.
Key result
Let k be a field, and let A and B be n ×n matrices with entries in k. Then the k[x]-modules (kn)A and (kn)B in Example 7.1(v) are k[x]-isomorphic if and only if there is a nonsingular matrix P with B = P AP−1.
Key result
If M and N are R-modules, where R is a commutative ring, then HomR(M, N) is an R-module, where addition has just been defined, and scalar multiplication is given by r f : m ↦f (rm). Moreover, there are distributive laws: If p: M′ →M and q : N →N ′, then ( f + g)p = f p + gp and q( f + g) = q f + qg for all f, g ∈HomR(M, N).
Key result
If f : M →N is an R-map of modules, then there is an R-isomorphism ϕ : M/ ker f →im f given by ϕ : m + ker f ↦f (m).
Key result
If T ⊆S ⊆M is a tower of submodules, then there is an R-isomorphism (M/T )/(S/T ) →M/S.
Key result
If T is a submodule of a module M, then there is a bijection ϕ : {intermediate submodules T ⊆S ⊆M} →{submodules of M/T } given by S ↦S/T. Moreover, S ⊆S′ in M if and only if S/T ⊆S′/T in M/T . Modules M S′ M/T S S′/T T S/T {0}
Key result
The following statements are equivalent for R-modules M, S, and T . (i) S ⊔T ∼= M. (ii) There exist injective R-maps i : S →M and j : T →M such that M = im i + im j and im i ∩im j = {0}. (iii) There exist R-maps i : S →M and j : T →M such that, for every m ∈M, there are unique s ∈S and t ∈T with m = is + jt. (iv) There are R-maps i : S →M, j : T →M, p: M →S, and q : M →T such that pi = 1S, qj = 1T , pj = 0, qi = 0, and ip + jq = 1M. Remark. The maps i and j are called injections, and the maps p and q are called projections. The equations pi = 1S and qj = 1T show that the maps i and j must be injective (so that im i ∼= S and im j ∼= T ) and the maps p and q must be surjective. 2Other common notations are S ⊕T and S × T . Modules
Key result
The following conditions are equivalent for an R-module M with submodules S and T . (i) M = S ⊕T . (ii) S + T = M and S ∩T = {0}. (iii) Each m ∈M has a unique expression of the form m = s + t for s ∈S and t ∈T .
Key result
A submodule S of a module M is a direct summand if and only if there exists a retraction ρ : M →S.
Key result
Let M = S1 + · · · + Sn, where the Si are submodules; that is, each m ∈M has a (not necessarily unique) expression of the form m = s1 + · · · + sn, where si ∈Si for all i. Then M = S1 ⊕· · · ⊕Sn if and only if, for each i, Si ∩⟨S1 + · · · + Si + · · · + Sn⟩= {0}, where Si means that the term Si is omitted from the sum.
Key result
(i) If 0 →A f→B g→C →0 is a short exact sequence, then A ∼= im f and B/ im f ∼= C. (ii) If T ⊆S ⊆M is a tower of submodules, then there is an exact sequence 0 →S/T f→M/T g→M/S →0.
Key result
(i) A commutative ring R is ascending-chain-finite if and only if every submodule of a finitely generated R-module M is itself finitely generated (ii) If R is a PID and if M can be generated by n elements, then every submodule of M can be generated by n or fewer elements. Remark.
Key result
is not true more generally. For example, if R is not a PID, there there is some ideal I that is not principal. Thus, R has one generator while its submodule I cannot be generated by one element. ◀
Key result
Let α ∈C and define Z[α] = { g(α): g(x) ∈Z[x] } . (i) Z[α] is a subring of C. Modules (ii) A complex number α is an algebraic integer if and only if Z[α] is a finitely generated additive abelian group. (iii) The set of all the algebraic integers is a subring of C.
Source-grounded examples
Worked source example
(i) Every vector space over a field k is a k-module. (ii) By the laws of exponents, Proposition 2.23, every abelian group is a Z-module. (iii) Every commutative ring R is a module over itself if we define scalar multiplication R × R →R to be the given multiplication of elements of R. More generally, every ideal I in R is an R-module, for if i ∈I and r ∈R, then ri ∈I. (iv) If S is a subring of a commutative ring R, then R is an S-module, where scalar multiplication S × R →R is just the given multiplication (s,r) ↦sr. For example, if k is a commutative ring, then k[X] is a k-module. (v) Let T : V →V be a linear transformation, where V is a finite-dimensional vector space over a field k. The vector space V can be made into a k[x]-module if scalar multiplication k[x] × V →V is defined as follows: If f (x) = m i=0 ci xi lies in k[x], then f (x)v = m i=0 ci xi v = m i=0 ciT i(v), where T 0 is the identity map 1V , and T i is the composite of T with itself i times if i ≥1. We denote V viewed as a k[x]-module by V T . Here is a special case of this construction. Let A be an n × n matrix with entries in k, and let T : kn →kn be the linear transformation T (w) = Aw, where w is an n ×1 column vector and Aw is matrix multiplication. Now the vector space kn is a k[x]-module if we define scalar multiplication k[x] × kn →kn as follows: If f (x) = m i=0 ci xi ∈k[x], then f (x)w = m i=0 ci xi w = m i=0 ci Aiw, where A0 = I is the identity matrix, and Ai is the ith power of A if i ≥1. We now show that (kn)T = (kn)A. Both modules are comprised of the same elements (namely, all n-tuples), and the scalar multiplications coincide: In (kn)T , we have xw = T (w); in (kn)A, we have xw = Aw; these are the same because T (w) = Aw. ◀ Here is the appropriate notion of homomorphism.
Worked source example
By Proposition 2.107, an abelian group G is simple if and only if G ∼= Ip for some prime p. ◀
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 modules, submodules and exact sequences?
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.