Ring extensions and endomorphism rings of a module (Q1316491)

All the rings are associative rings with identity element and all the modules are unitary. The authors investigate the following two conditions: (T) if \(R \supset S\) is a ring extension and \(M\) a right \(R\)- module then \(M \otimes_ S R\) is a direct summand of a finite direct sum of copies of \(M\) as a right \(R\)-module; (H) if \(P \supset Q\) is a ring extension and \(M\) a left \(P\)-module then \(\text{Hom}({_ QP},{_ QM})\) is a direct summand of a finite direct sum of copies of \(M\) as a left \(P\)-module. In the first paragraph the relations between these two conditions are treated in the case when \(P = \text{End} (M_ S)\), \(Q = \text{End}(M_ R)\) and \(R = \text{End}(_ QM)\), \(S = \text{End}(_ PM)\). The results obtained are applied in the next part to \(H\)-separable extensions of rings. The last item is devoted to the properties of relative generators and cogenerators.