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 an initial object. However, it is not a terminal object: in fact, the only rings with homomorphisms into the trivial rings are themselves trivial, as we require ring homomorphisms to preserve 0 and 1.

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

See also

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