Remarques sur les modules S-pur-projectifs. (Remarks on the S-pure- projective modules) (Q1109856)

By A is denoted an infinite connected artinian algebra. All modules are left A-modules. Let \(S=(M_ i)_{i\in I}\) be a system of finitely generated indecomposable A-modules pairwise disjoint. An A-module is said to be S-pure-projective (S.P.P.) if it is isomorphic to a direct sum of objects of S. The author shows that the following statements are equivalent: (1) I is finite; (2) Every module elementary equivalent with an S.P.P. module is S.P.P.; (3) Every ultrapower of S.P.P. modules is S.P.P.; (4) Every ultraproduct of S.P.P. modules is S.P.P.; (5) Every ultraproduct of S.P.P. modules has a finitely generated indecomposable direct summand; (6) Every ultraproduct of objects of S has a finitely generated indecomposable direct summand.