---
tag: DTWDS6KAKN
kind: definition
name: Left ideal
msc: [16D25]
defines: 5N2CJZ47D4
uses: [0894J980RK, G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_(ring_theory)
---

## Definition

Let $R$ be a [[ring|0894J980RK]].
A *left ideal* of $R$ is a subset $I \subseteq R$ such that:

1. $I$ is a subgroup of the additive [[group|G5S1M9]] $(R, +)$.
2. $ra \in I$ for all $r \in R$ and $a \in I$.

## Remarks

A *right ideal* is defined in the same way with $ar \in I$ in clause 2.
A [[two-sided ideal|0DDA6CY7GJ]] is a subset that is both a left and a right ideal.
Example: let $K$ be a field and let $I \subseteq M_2(K)$ be the set of matrices whose second column is zero.
It is an additive subgroup, and for $A \in M_2(K)$ and $X \in I$ the second column of $AX$ is $A$ times the second column of $X$, which is zero.
So $I$ is a left ideal.
Non-example: the set $J$ of matrices whose second row is zero is an additive subgroup, but $E_{21} E_{11} = E_{21} \notin J$, where $E_{ij}$ is the matrix with $1$ in position $(i,j)$ and $0$ elsewhere.
So clause 2 fails for $J$; it is a right ideal instead.
The left ideals of $R$ are the submodules of $R$ as a left module over itself.
For each $a \in R$, the set $Ra = \set{ra : r \in R}$ is the left ideal generated by $a$.