Divisibility in a commutative ring

Definition

Let $R$ be a commutative ring and let $a, b \in R$. The element $a$ divides $b$, written $a \mid b$, if there is $c \in R$ with $b = a c$. Then $a$ is a divisor of $b$, and $b$ is a multiple of $a$.

Remarks

The relation $a \mid b$ holds if and only if $(b) \subseteq (a)$ for the principal ideals. Every element divides $0$, and $0$ divides only $0$. A unit divides every element. In $\mathbb{Z}[i]$, $1 + i$ divides $2$, because $2 = (1 + i)(1 - i)$.

Explanations

L1 human

Divisibility in a ring works as for integers: $a$ divides $b$ if $b$ is $a$ times some element of the ring. The answer depends on the ring: $2$ does not divide $1$ in $\mathbb{Z}$, but it does in $\mathbb{Q}$.