I already proved that if i was given a surjective ring morphism f from R to S and then if a∈R is invertible, central, idempotent, or nilpotent, respectively then f(a) also is. But im looking for counterexample to show the reciprocal is not always true. Thanks
2026-03-30 14:59:03.1774882743
Counterexamples for the image of central,idempotent,invertible and nilpotent elements of a ring
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Try $R=\Bbb Z$ for invertible, idempotent, nilpotent.
Can you find a noncommutative ring with a commutative quotient? It's not hard, but you may be unfamiliar with examples of or construction methods for noncommutative rings.