Zero ring

A zero ring is a ring with one element, 0 (equal to 1), with the additive and multiplicative structure of the trivial group. Technically speaking, there are infinitely many zero rings (one for each possible element "0"), but they are all trivially isomorphic, so by abuse of language we may refer to the zero ring.

Proposition. If $R$ is a ring in which $0=1$, then $R$ is a trivial ring.

Proof. For any $x \in R$, we have $$x = 1 \cdot x = 0 \cdot x = 0 . \qquad \blacksquare$$

In the category of rings, the zero ring is a terminal object, through the trivial ring homomorphism. However, it is not an initial object. This can be seen by the fact that ring homomorphisms must preserve the identities. Clearly a ring homomorphism, $\varphi(1_0)$ where $\varphi : \mathbf{0} \to R$ cannot be defined as both $1_R$ and $0_R$ since AoPS defines rings to have multiplicative identity (some sources vary here). It can instead be shown that the integers, $\mathbb{Z}$ is initial in $\mathbf{Ring}$.

Note that by convention there is no "trivial field" or "zero field", as we usually require 0 and 1 to be distinct in fields.