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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?

gen
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    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 – Maxime Ramzi May 29 '19 at 10:38
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    In a semisimple category all sequences split so all objects are injective and projective. – Ben May 29 '19 at 10:40
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    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*. – rschwieb May 29 '19 at 10:42

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