Zero ring

Example

The zero ring is the set $\{0\}$ with the operations $0 + 0 = 0$ and $0 \cdot 0 = 0$. It is a commutative ring in which $1 = 0$.

Remarks

A ring $R$ is the zero ring if and only if $1 = 0$ in $R$. If $1 = 0$, then $a = a \cdot 1 = a \cdot 0 = 0$ for each $a \in R$. The zero ring is not an integral domain and not a field, because these definitions require $1 \neq 0$.

Explanations

L1 human

The zero ring has only one element. All the axioms of a ring hold, but in a trivial way. It is the only ring in which $1$ and $0$ are the same element.