---
tag: 0MBJVDSK1R
kind: concept
name: Sum of ideals
msc: [13A15]
uses: [0MYSA6MYNA]
sources:
  - url: https://en.wikipedia.org/wiki/Ideal_(ring_theory)
    locator: Ideal operations
---

## Summary

Let $R$ be a [[ring|0894J980RK]] and let $I$ and $J$ be left [[ideals|0MYSA6MYNA]] of $R$.
The sum of $I$ and $J$ is $I + J = \{a + b : a \in I, b \in J\}$.
It is a left ideal, and it is the smallest left ideal that contains $I$ and $J$.

## Remarks

The sum $I + J$ is the ideal [[generated by|030TZ0DQWT]] $I \cup J$.
In $\mathbb{Z}$, $m\mathbb{Z} + n\mathbb{Z} = d\mathbb{Z}$, where $d$ is the greatest common divisor of $m$ and $n$.
The ideals $I$ and $J$ are [[coprime|0MDJP82TEH]] when $I + J = R$.

## Explanations

### L1 human

To add two ideals, take all the sums of one element from each.
In the integers, the sum of the multiples of $4$ and the multiples of $6$ is the set of the multiples of $2$.