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In the category of R-modules an object that is both injective and projective is necessarily the zero module.

Are there any abelian categories with examples non-zero objects that are both injective and projective?

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    $\begingroup$ Well that's not true if $R$ is a field for starters, or not a domain in general : if $k$ is a field and $G$ a finite group, then $k[G]$-injective and $k[G]$-projective mean the same thing $\endgroup$ – Max May 29 at 10:38
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    $\begingroup$ In a semisimple category all sequences split so all objects are injective and projective. $\endgroup$ – Ben May 29 at 10:40
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    $\begingroup$ semisimple rings are characterized by having all modules projective and injective. Quasi-frobenius rings are characterized by the fact their injective modules are precisely their projective modules. So the proposition you’re citing isn’t true in general. Notice your link is talking about nonfield domains. $\endgroup$ – rschwieb May 29 at 10:42

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