Abstract
A module M is said to satisfy the C
12 condition if every submodule of M is essentially embedded in a direct summand of M. It is known that the C
11 (and hence also C
1) condition implies the C
12 condition. We show that the class of C
12-modules is closed under direct sums and also essential extensions whenever any module in the class is relative injective with respect to its essential extensions. We prove that if M is a -module with cancellable socle and satisfies ascending chain (respectively, descending chain) condition on essential submodules, then M is a direct sum of a semisimple and a Noetherian (respectively, Artinian) submodules. Moreover, a C
12-module with cancellable socle is shown to be a direct sum of a module with essential socle and a module with zero socle. An example is constructed to show that the reverse of the last result do not hold.
2000 Mathematics Subject Classification:
Notes
Communicated by T. Albu.