Let R R be a commutative semilocal noetherian ring, Λ \Lambda a left noetherian R R -algebra and M , N M,N finitely generated left Λ \Lambda -modules such that End Λ ( M ) {\operatorname {End} _\Lambda }(M) is of finite type over R R . By R ^ \hat R we denote the ( rad ⁡ R ) (\operatorname {rad} R) -adic completion of R R . Theorem. M M is Λ \Lambda -isomorphic to a direct summand of N N iff R ^ ⊗ R M \hat R{ \otimes _R}M is R ^ ⊗ R Λ \hat R{ \otimes _R}\Lambda -isomorphic to a direct summand of R ^ ⊗ R N \hat R{ \otimes _R}N . This result is used to prove a generalization of the Noether-Deuring theorem. Let S S be a commutative R R -algebra which is a faithful projective R R -module of finite type; then M M is Λ \Lambda -isomorphic to direct summand of N N iff S ⊗ R M S{ \otimes _R}M is S ⊗ R Λ S{ \otimes _R}\Lambda -isomorphic to a direct summand of S ⊗ R N S{ \otimes _R}N .