5th International Conference Noncommutative Rings and their Applications, Lens, France, 12 - 15 June 2017, vol.727, pp.223-236
This paper introduces the notion of essentially ADS (e-ADS) modules. Basic structural properties and examples of e-ADS modules are presented. In particular, it is proved that (1) The class of all e-ADS modules properly contains all ADS as well as automorphism invariant modules. e-ADS modules serves also as a tool for characterization of various classes of rings. It is shown that: (2) R is a QF-ring if and only if every projective right R-module is e-ADS; (3) R is a semisimple Artinian ring if and only if every e-ADS module is injective. The final part of this paper describes properties of e-ADS rings, which allow to prove a criterion of e-ADS modules for non-singular rings: (4) Let R be a right non-singular ring and Q be its the right maximal ring of quotients. Then R is a right e-ADS ring if and only if either eQ not congruent to (1 - e) Q for any idempotent e is an element of R or R congruent to M-2(A) for a suitable right automorphism invariant ring A.