I'm asking this because I could only think of infinite rings where this is true. This must include rings.
2026-02-23 19:18:49.1771874329
Find a finite ring with elements other than the zero element, units, or zero-divisors.
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There's a reason why you can't find one! If by "ring" you mean a ring with unity, then every non-zero element of a finite ring $R$ is either a unit or a zero divisor. See this link for a proof.