---
tag: 5N2CJZ47D4
kind: concept
name: Left ideal
msc: [16D25]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_(ring_theory)
---

## Summary

A left ideal of a [[ring|0B1261B5ED]] $R$ is an additive subgroup that is closed under multiplication on the left by every element of $R$.
Left ideals are exactly the submodules of $R$ viewed as a left module over itself.
In a commutative ring left ideals, right ideals and [[two-sided ideals|0CD87SFM2N]] all coincide with [[ideals|096Z9MP2ME]].
In a noncommutative ring the three notions differ, and left ideals carry much of the module theory of $R$.

## Explanations

### L1 human

A left ideal is a part of a ring that absorbs multiplication from the left: multiply one of its elements on the left by anything, and the product stays inside.
Multiplication on the right may lead out of it.
In the ring of $2 \times 2$ matrices, the matrices whose second column is zero form a left ideal.
Multiplying such a matrix on the left by any matrix keeps the second column zero.
Multiplying it on the right can fill the second column, so it is not a right ideal.