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It is well-known that if R is a Noetherian ring, and every finitely generated right R-module embeds in projective, then R is a self-injective. My question is that could one replace "finitely generated" by "cyclic"?

Thank you in advance for your answer.

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  • $\begingroup$ Cyclic is finitely generated (in other terminologies, one says "monogeneous"). It means "with one generator". $\endgroup$
    – Duchamp Gérard H. E.
    Commented Aug 22, 2018 at 12:52
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    $\begingroup$ Yes, but the question is" If R is right Noetherian ring and every right cyclic R-module embeds in projective, then R is a self-injective"? $\endgroup$
    – R. Shhaied
    Commented Aug 22, 2018 at 13:40
  • $\begingroup$ Ok, I understand $\endgroup$
    – Duchamp Gérard H. E.
    Commented Aug 22, 2018 at 15:58
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    $\begingroup$ Do you happen to have a reference that I could take a look at for the 'well-known' result? Thanks $\endgroup$
    – rschwieb
    Commented Aug 22, 2018 at 16:16
  • $\begingroup$ www.researchgate.net/publication/225714368_Embedding_modules_in_projectives_A_report_on_a_problem $\endgroup$
    – R. Shhaied
    Commented Aug 23, 2018 at 0:05

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