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Modules, Submodules, Quotient Modules and the Isomorphism Theorems: Examples and Counterexamples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-03Open item page →

The residue classes Z/n\mathbb Z/n as a Z\mathbb Z-module

Example

For n≥1n\ge1, the additive group of residue classes Z/n\mathbb Z/n is a left Z\mathbb Z-module under

r [a]n:=[ra]n.r\,[a]_n:=[ra]_n.

Facts & Assumptions

Given: An integer n≥1n\ge1, a scalar r∈Zr\in\mathbb Z, and residue classes [a]n,[b]n∈Z/n[a]_n,[b]_n\in\mathbb Z/n.

[L1]

Addition and multiplication of residue classes are representative-independent (Addition and multiplication on Z/n\mathbb{Z}/n by [a]n+[b]n=[a+b]n[a]_n+[b]_n=[a+b]_n and [a]n[b]n=[ab]n[a]_n[b]_n=[ab]_n).

[L2]

The integers form a commutative ring (The integers form a commutative ring).

[L3]

The additive residue classes form an abelian group (For every natural nn, (Z/n,+)(\mathbb{Z}/n,+) is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).

[L4]

A left module is an abelian group with a unital distributive associative scalar action (Unital left and right modules over a ring; unqualified module means left module).

Verification

technique · direct
1.1

The displayed action is representative-independent: replacing aa by a congruent integer leaves rara congruent modulo nn.

L1L2given
2.1

Integer distributivity and associativity give r([a]n+[b]n)=r[a+b]n=[ra+rb]n=r[a]n+r[b]nr([a]_n+[b]_n)=r[a+b]_n=[ra+rb]_n=r[a]_n+r[b]_n, (r+s)[a]n=r[a]n+s[a]n(r+s)[a]_n=r[a]_n+s[a]_n, and (rs)[a]n=r(s[a]n)(rs)[a]_n=r(s[a]_n).

step 1.1L1L2given
3.1

Finally 1[a]n=[a]n1[a]_n=[a]_n, and [L3] supplies the abelian additive group; therefore the module axioms hold.

step 2.1L1L2L3L4∎
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-03Open item page →

Left ideals are exactly the submodules of the regular left module RR_RR

Example

Let RR be a ring. On its additive group define the left scalar action r⋅x:=rxr\cdot x:=rx, using multiplication in RR. This makes RR the regular left module, written RR_RR. A subset I⊆RI\subseteq R is a submodule of RR_RR if and only if II is a left ideal of RR.

Facts & Assumptions

Given: A ring RR and its underlying additive group.

[L1]

A left module is an abelian group with a unital scalar action satisfying the two distributive laws and associativity of scalar multiplication (Unital left and right modules over a ring; unqualified module means left module).

[L2]

A subset is a submodule exactly when it is an additive subgroup and is closed under multiplication by every scalar (Submodule of a module).

[L3]

A left ideal is an additive subgroup II such that ri∈Iri\in I for all r∈Rr\in R and i∈Ii\in I (Left, right and two-sided ideals).

[L4]

In a ring, addition is an abelian-group operation, multiplication is associative and unital, and multiplication distributes over addition on both sides (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides).

Proof

technique · direct
1.1

The additive group of RR is abelian. For r,s,x,y∈Rr,s,x,y\in R, the ring laws give r(x+y)=rx+ryr(x+y)=rx+ry, (r+s)x=rx+sx(r+s)x=rx+sx, (rs)x=r(sx)(rs)x=r(sx), and 1Rx=x1_Rx=x. Thus r⋅x:=rxr\cdot x:=rx makes RR a left RR-module.

L1L4given
2.1

If II is a left ideal, then it is an additive subgroup and ri∈Iri\in I for all r∈Rr\in R, i∈Ii\in I. Hence it is a submodule of RR_RR.

step 1.1L2L3
2.2

Conversely, if II is a submodule of RR_RR, then it is an additive subgroup and is closed under the scalar action, which here says exactly that ri∈Iri\in I for all r∈Rr\in R, i∈Ii\in I. Hence II is a left ideal.

step 1.1L2L3
3.1

Steps 2.1 and 2.2 prove the claimed equivalence.

step 2.1step 2.2∎

Remarks

  • With the right regular module RRR_R, the same argument identifies its submodules with the right ideals. Two-sided ideals are precisely the subsets that are submodules in both regular-module structures.

Sources

Standard references

Recommended treatments; not extraction sources.