Geodesic
Pure-semisimplicity is preserved under elementary equivalence
Glasgow mathematical journal, Tome 36 (1994) no. 3, pp. 345-346

Voir la notice de l'article provenant de la source Cambridge University Press

In the present note, Σr denotes the class of all right pure semisimple rings (= right pure global dimension zero). It is known that if R ∈ Σr, then R is right artinian and every indecomposable right R-module is finitely generated. The class Σr is not closed under ultraproducts [4]. While Σr is closed under elementary descent (i.e. if S ∈ Σr and R is an elementary subring of S then R ∈ σr) [4], it is an open question whether right pure-semisimplicity is preserved under the passage to ultrapowers [4, Prob. 11.16]. In this note, this question is answered in the affirmative. 
Zayed, Maher. Pure-semisimplicity is preserved under elementary equivalence. Glasgow mathematical journal, Tome 36 (1994) no. 3, pp. 345-346. doi: 10.1017/S0017089500030949
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