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Give a nontrivial example of an artin algebra $R$ such that $R$ is pure-injective as an $R$-module.

Clearly $0$-Gorenstein (self-injective) artin algebra has this property. Can anyone give me a nontrivial example? Thanks!

Edit. R. Goebel, J. Trlifaj's book Approximations and Endomorphism Algebras of Modules in Corollary 1.2.22 says over an artin algebra $R$, $R$ is always pure-injective as a left $R$-module. Thanks for all helpful discussions.

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According to Corollary 4.2 of Leif Melkersson's paper Cohomological Properties of Modules with Secondary Representations, any Artinian module over a commutative Noetherian ring is pure injective. In particular, any commutative Artinian ring is pure injective over itself, since Artinian rings are Noetherian by the Hopkins-Levitzki theorem.

An example which is not self-injective is given by $K[x,y]/(x^2, y^2, xy)$, for any field $K$.

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    $\begingroup$ Nice! For those who don't have access to the paper: the claim is reduced to the case of a complete local ring first, where by Matlis duality any Artinian module is of the form $\text{Hom}(M,E)$ for $M$ a finitely generated module and $E$ the injective hull of the residue field. Now it follows directly from the definition of pure injectivity that firstly any injective is pure-injective and secondly that with any pure-injective $I$ also $\text{Hom}(N,I)$ is pure-injective for any module $N$. $\endgroup$
    – Hanno
    Commented Jan 7, 2015 at 15:46
  • $\begingroup$ @Hanno Thank you for this nice summary! I hadn't realized that the access to the paper was restricted. $\endgroup$
    – Pierre-Guy Plamondon
    Commented Jan 7, 2015 at 15:48
  • $\begingroup$ There is no restriction to that paper. $\endgroup$
    – user26857
    Commented Jan 7, 2015 at 18:56

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