# Why is this right principally injective ring a self-injective ring?

If R is semiprime, 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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Quick question: what do you mean by "right principally injective" - right principal ideals are injective ? (surely not ?!) Kind regards - Stephan F. Kroneck. –  bonnbaki Sep 8 '11 at 13:46
Yes,and principally injective means divisible.From Lam:Lectures on modules and rings,p119. –  Strongart Sep 10 '11 at 6:01
See if you can use the chain condition to show that $R$ is equal to $Q_{max}^r(R)$, since $R$ already embeds in $Q_{max}^r(R)$. I'd also like to add that I've seen that the ACC on all right ideals with $R$ right princ-inj means $R$ is perfect, but I can't seem to find anything with just point annihilators. Another thing is that I hope you didn't switch up sides, because they can be tricky, here. –  rschwieb Apr 29 '12 at 20:16
Right principally injective rings are those for 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. –  rschwieb Apr 30 '12 at 1:35