Sum of ideals

Summary

Let $R$ be a ring and let $I$ and $J$ be left ideals 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 $I \cup J$. 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 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 $2$.