nLab
trivial ring

The trivial ring, or zero ring, is the ring with a single element, which is both $0$ and $1$. We usually denote the trivial ring as $0$ or $\{0\}$, even though $1$ or $\{1\}$ would make as much sense. It is the only ring in which $0=1$, by the proof

$$x=1x=0x=0.$$x = 1 x = 0 x = 0 .

The trivial ring is the terminal object in Ring. It is both terminal and initial (hence a zero object) in the category of nonunital rings, but it is not initial in $\mathrm{Ring}$ itself (defined as the category of unital rings and unital ring homomorphisms). In fact, there are no unital ring homomorphisms from the trivial ring to any nontrivial ring!

The trivial ring is an example of a trivial algebra.