ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-03

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)

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