A note on the stability of pure-injective modules

Abstract

ABSTRACTLet R be a ring. Then a left R-module N is pure-injective if and only if HomR(M,N) is a pure-injective left S-module for any ring S and any (R,S)-bimodule . If R is a commutative ring and M,N are R-modules with N pure-injective, then is a pure-injective R-module for any n≥0. Let R and S be rings and let be an (S,R)-bimodule and M a finitely presented left R-module. If N is pure-injective as a left S-module, then the left S-module N⊗RM is pure-injective; and if R is left coherent, then the left S-module is pure-injective for any n≥1.