Let $R$ be a ring with identity whose (right) socle $S$ contains its nilpotent elements. Is it necessarily true that the quotient $R/S$ is an abelian ring? ( By an abelian ring I mean a ring whose idempotents are central.) For example, $R$ may be a reduced ring -such as a domain-which does trivialy satisfies the hypothesis. Thanks for any answer!
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