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If $R$ is a semiprime ring, right principally injective and satisfies ACC on right annihilators of elements, is it self-injective?

I only know that it is right nonsingular.

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    Right principally injective rings are those $f$or which a module homomorphism from a principal right ideal into the ring can be extended to the entire ring. Another characterization is that l(r(a))=Ra for every a∈R , where l denotes the left annihilator and r the right annihilator.2012-04-30

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