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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 $Z / n$ as a $Z$ -module

Example

For $n \geq 1$ , the additive group of residue classes $Z / n$ is a left $Z$ -module under

$r [a]_{n} : = [r a]_{n} .$

Facts & Assumptions

Given: An integer $n \geq 1$ , a scalar $r \in Z$ , and residue classes $[a]_{n}, [b]_{n} \in Z / n$ .

[L1]

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

[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 $n$ , $(Z / n, +)$ 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 $a$ by a congruent integer leaves $r a$ congruent modulo $n$ .

L1L2given

2.1

Integer distributivity and associativity give $r ([a]_{n} + [b]_{n}) = r [a + b]_{n} = [r a + r b]_{n} = r [a]_{n} + r [b]_{n}$ , $(r + s) [a]_{n} = r [a]_{n} + s [a]_{n}$ , and $(r s) [a]_{n} = r (s [a]_{n})$ .

step 1.1L1L2given

3.1

Finally $1 [a]_{n} = [a]_{n}$ , 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 $_{R} R$

Example

Let $R$ be a ring. On its additive group define the left scalar action $r \cdot x : = r x$ , using multiplication in $R$ . This makes $R$ the regular left module, written $_{R} R$ . A subset $I \subseteq R$ is a submodule of $_{R} R$ if and only if $I$ is a left ideal of $R$ .

Facts & Assumptions

Given: A ring $R$ 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 $I$ such that $r i \in I$ for all $r \in R$ and $i \in I$ (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 $R$ is abelian. For $r, s, x, y \in R$ , the ring laws give $r (x + y) = r x + r y$ , $(r + s) x = r x + s x$ , $(r s) x = r (s x)$ , and $1_{R} x = x$ . Thus $r \cdot x : = r x$ makes $R$ a left $R$ -module.

L1L4given

2.1

If $I$ is a left ideal, then it is an additive subgroup and $r i \in I$ for all $r \in R$ , $i \in I$ . Hence it is a submodule of $_{R} R$ .

step 1.1L2L3

2.2

Conversely, if $I$ is a submodule of $_{R} R$ , then it is an additive subgroup and is closed under the scalar action, which here says exactly that $r i \in I$ for all $r \in R$ , $i \in I$ . Hence $I$ 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 $R_{R}$ , 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

McGerty, Algebra II: Rings and Modules, Section 3

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

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

The residue classes Z/nZ/n as a ZZ-module

Example

Facts & Assumptions

Verification

Left ideals are exactly the submodules of the regular left module RRR​R

Example

Facts & Assumptions

Proof

Remarks

Sources

The residue classes $Z / n$ as a $Z$ -module

Left ideals are exactly the submodules of the regular left module $_{R} R$