Composition series relative to a module (Q800462)

The results of \textit{O. Goldman} [J. Algebra 35, 308-341 (1975; Zbl 0313.16002)] for composition series relative to an hereditary torsion theory are generalized to the non-hereditary case. In particular, for a module \(U_ R\), a module \(M_ R\) is U-torsion if \(Hom_ R(M_ R,U_ R)=0\); \(N_ R\) is U-cocritical if \(N_ R\) is U-torsionless and any proper homomorphic image of \(N_ R\) is U-torsion. Then a U-composition series can be defined in the obvious way, and the analogs for the usual existence, refinement, and uniqueness theorems are proved. If U is quasi- injective and \(S=End_ R(U)\), then there is a natural Galois type relationship between a U-composition series for \(M_ R\) and a composition series of \({}_ SHom_ R(M_ R,U_ R)=_ SM^*\). Applications are made to the study of endomorphism rings and the structure of \({}_ SM^*\).