5th International Conference Noncommutative Rings and their Applications, Lens, France, 12 - 15 June 2017, vol.727, pp.211-221
Let R be a ring and let M be an R-module with S = End(R)(M). Then M is called d-retractable if for U <= V v M, l(S)(U)/l(S)(V) << S-S/l(S)(V) implies that V/U << M/U. We prove that every nonzero d-retractable module is coretractable but the converse is not true, in general. It is shown that the R-module R-R is d-retractable if and only if R is a semisimple ring. We show that every quasi-injective d-retractable module with essential socle is semisimple. The class of rings R for which every finitely cogenerated R-module is d-retractable, is shown to be exactly that of the right V-rings. Some relevant counterexamples are indicated.