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Left ideals are exactly the submodules of the regular left module $_RR$

Example

Let $R$ be a ring. On its additive group define the left scalar action $r\cdot x:=rx$ , using multiplication in $R$ . This makes $R$ the regular left module, written $_RR$ . A subset $I\subseteq R$ is a submodule of $_RR$ 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 $ri\in I$ 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 $r(x+y)=rx+ry$ , $(r+s)x=rx+sx$ , $(rs)x=r(sx)$ , and $1_Rx=x$ . Thus $r\cdot x:=rx$ makes $R$ a left $R$ -module.

L1L4given

2.1

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

step 1.1L2L3

2.2

Conversely, if $I$ is a submodule of $_RR$ , then it is an additive subgroup and is closed under the scalar action, which here says exactly that $ri\in I$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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 RR_RRR​R