The rational Loewy series and nilpotent ideals of endomorphism rings (Q1099235)

Let R be a ring, let L, M be left R-modules, let QE(M) be the quasi- injective hull of M, let S be the endomorphism ring of M, and let \(N(S)=\{f\in S|\) ker f is essential in \(M\}\). Then M is called L- distinguished if, for each nonzero homomorphism \(f: X\to M\), there exists a homomorphism \(g: L\to X\) such that fg\(\neq 0\). A submodule X of L is QE(M)-closed if L/X is cogenerated by QE(M). Let \(\sigma\) [M] be the class of all modules that are isomorphic to submodules of M-generated modules. If there exists L such that M is L-distinguished in \(\sigma\) [M] and L satisfies the DCC on QE(M)-closed submodules, then M has a finite rational Loewy series and N(S) is nilpotent. Many nice applications of this result are made to the nilpotency of ideals in S and to the nilpotency of nil subrings of S. An important feature is that, unlike many previous results, M need not be a \(\Delta\)-module in order to apply these results. If M has DCC on rationally closed submodules, then nil subrings of S are nilpotent. The latter part of the paper examines the index of nilpotency of N(S). A bound is given in terms of linkage theory. Under the assumption that M is quasi-injective, the index of nilpotency is equal to the length of the rational Loewy series in many cases.