Modules for which every submodule has a supplement that is a direct summand

Abstract EN

Let R be a ring and M an R-module.

We say that M satisfies the property (Dn) if every submodule of M has a supplement which is a direct summand of M.

In this paper we prove that for a commutative ring R the following are equivalent: (/) Every R-module satisfies (D,,); (ii) R is an artinian principal ideal ring.

Moreover we prove that a module M over a commutative noetherian ring satisfies (Dn) if and only if M=P(M) ® L, for some (coatomic) submodule L ofM and P(M) and L both satisfy (Dn), here P(M) is the sum of all radical submodules of M.

We also give some results concerning modules satisfying (D11).