COMMUNICATIONS IN ALGEBRA, vol.41, no.1, pp.83-98, 2013 (SCI-Expanded)
Let R be an arbitrary ring. A nonzero unital right R-module M is called a second module if M and all its nonzero homomorphic images have the same annihilator in R. It is proved that if R is a ring such that R/P is a left bounded left Goldie ring for every prime ideal P of R, then a right R-module M is a second module if and only if Q = ann(R)(M) is a prime ideal of R and M is a divisible right (R/Q)-module. If a ring R satisfies the ascending chain condition on two-sided ideals, then every nonzero R-module has a nonzero homomorphic image which is a second module. Every nonzero Artinian module contains second submodules and there are only a finite number of maximal members in the collection of second submodules. If R is a ring and M is a nonzero right R-module such that M contains a proper submodule N with M/N a second module and M has finite hollow dimension n, for some positive integer n, then there exist a positive integer k = n and prime ideals P-i(1 <= i <= k) such that if L is a proper submodule of M with M/L a second module, then M/L has annihilator Pi for some 1 <= i <= k. Every second submodule of an Artinian module is a finite sum of hollow second submodules.