Coprime ideals

Definition

Let $R$ be a commutative ring and let $I$ and $J$ be ideals of $R$. The ideals $I$ and $J$ are coprime if $I + J = R$, where $I + J$ is the sum of the ideals. Equivalently, there are $a \in I$ and $b \in J$ with $a + b = 1$.

Remarks

In $\mathbb{Z}$, $m\mathbb{Z} + n\mathbb{Z} = \gcd(m, n)\mathbb{Z}$. So $m\mathbb{Z}$ and $n\mathbb{Z}$ are coprime if and only if $\gcd(m, n) = 1$. If $I$ and $J$ are coprime, then $I \cap J = I J$. Two distinct maximal ideals are always coprime.

Explanations

L1 human

For integers, $3$ and $5$ are coprime, and $2 \cdot 3 - 1 \cdot 5 = 1$. So $1$ is a multiple of $3$ plus a multiple of $5$. Coprime ideals generalize this: $1$ is an element of $I$ plus an element of $J$.