Modules, Algebras and Module Homomorphisms
Handbook guide to modules, algebras and module homomorphisms with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Modules and Algebras
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.
The Isomorphism Theorems For Modules
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.
Definitions and Comments
A vector space M over a field R is a set of objects called vectors, which can be added, subtracted and multiplied by scalars (members of the underlying field). Thus M is an abelian group under addition, and for each r ∈R and x ∈M one has an element rx ∈M. Scalar multiplication is distributive and associative, and the multiplicative identity of the field acts as an identity on vectors. Formally, r(x + y) = rx + ry; (r + s)x = rx + sx; r(sx) = (rs)x; 1x = x for all x, y ∈M and r, s ∈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.
Some Basic Properties of Modules
Some Basic Properties of Modules Let M be an R-module. The technique given for rings in (2.1.1) can be applied to establish the following results, which hold for any x ∈M and r ∈R. We distinguish the zero vector 0M from the zero scalar 0R. (1) r0M = 0M [r0M = r(0M + 0M) = r0M + r0M] (2) 0Rx = 0M [0Rx = (0R + 0R)x = 0Rx + 0Rx] (3) (−r)x = r(−x) = −(rx) [as in (2) of (2.1.1) with a replaced by r and b by x] (4) If R is a field, or more generally a division ring, then rx = 0M implies that either r = 0R or x = 0M. [If r ̸= 0, multiply the equation rx = 0M by r−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.
Examples
1. If M is a vector space over the field R, then M is an R-module. 2. Any ring R is a module over itself. Rather than check all the formal requirements, think intuitively: Elements of a ring can be added and subtracted, and one can certainly multiply r ∈R by x ∈R, and the usual rules of arithmetic apply. 3.
Proof / verification strategy: Treat this as an illustration of the surrounding definitions. Recompute the stated relations directly and keep the numerical or structural choices local to the example.
Definition
Let R be a commutative ring. Call M is an algebra over R, or that M is an R-algebra, if M is an R-module that is also a ring (not necessarily commutative), and the ring and module operations are compatible, i.e., r(xy) = (rx)y = x(ry) for all x, y ∈M and 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.
Examples
1. Every commutative ring R is an algebra over itself (see Example 2 of (4.1.3)). 2. An arbitrary ring R is always a Z-algebra (see Example 5 of (4.1.3)). 3. If R is a commutative ring, then Mn(R), the set of all n × n matrices with entries in R, is an R-algebra (see Example 4 of (4.1.3)). 4.
Proof / verification strategy: Treat this as an illustration of the surrounding definitions. Recompute the stated relations directly and keep the numerical or structural choices local to the example.
Definition
If N is a nonempty subset of the R-module M, call N is a submodule of M (notation N ≤M) if for every x, y ∈N and r, s ∈R, one has rx + sy ∈N. If M is an R-algebra, call N is a subalgebra if N is a submodule that is also a subring. For example, if A is an abelian group (= Z-module), the submodules of A are the subsets closed under addition and multiplication by an integer (which amounts to addition also). Thus the submodules of A are simply the subgroups.
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.
Definition
Let M and N be R-modules. A module homomorphism (also called an R-homomorphism) from M to N is a map f : M →N such that f(rx + sy) = rf(x) + sf(y) for all x, y ∈M and r, s ∈R. Equivalently, f(x + y) = f(x) + f(y) and f(rx) = rf(x) for all x, y ∈M and r ∈R. The kernel of a homomorphism f is ker f = {x ∈M : f(x) = 0}, and the image of f is {f(x) : x ∈M}.
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.
Another Way to Describe an Algebra
Another Way to Describe an Algebra Assume that A is an algebra over the commutative ring R, and consider the map r →r1 of R into A. The commutativity of R and the compatibility of the ring and module operations imply that the map is a ring homomorphism.
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.
Factor Theorem For Modules Any module homomorphism f : M →M ′ whose
Factor Theorem For Modules Any module homomorphism f : M →M ′ whose kernel contains N can be factored through M/N. Equivalently, there is a unique module homomorphism f : M/N →M ′ such that f(x + N) = f(x). Furthermore, (i) f is an epimorphism if and only if f is an epimorphism; (ii) f is a monomorphism if and only if ker f = N; (iii) f is an isomorphism if and only if f is a epimorphism and kerf = N.
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Theorem
First Isomorphism Theorem For Modules If f : M →M ′ is a module homomorphism with kernel N, then the image of f is isomorphic to M/N.
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Second Isomorphism Theorem For Modules Let S and T be submodules of M,
Second Isomorphism Theorem For Modules Let S and T be submodules of M, and let S + T = {x + y : x ∈S, y ∈T}. Then S + T and S ∩T are submodules of M and (S + T)/T ∼= S/(S ∩T).
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Third Isomorphism Theorem For Modules If N ≤L ≤M, then
Third Isomorphism Theorem For Modules If N ≤L ≤M, then M/L ∼= (M/N)/(L/N).
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Correspondence Theorem For Modules Let N be a submodule of the R-module
Correspondence Theorem For Modules Let N be a submodule of the R-module M. The map S →S/N sets up a one-to-one correspondence between the set of all submodules of M containing N and the set of all submodules of M/N. The inverse of the map is T →π−1(T), where π is the canonical map: M →M/N.
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.
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 source exercises reinforce definition checking, construction of examples and direct use of the formal results in this article.
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 |
|---|---|---|
| 4.1 | Modules and Algebras | 65–67 |
| 4.2 | The Isomorphism Theorems For Modules | 68–68 |
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.