On lifting and extending properties on direct sums of hollow uniform modules

Abstract

A module M is said to belifting if, for anysubmodule N of M, there exists a direct summand X of M contained in N such that N/X is small in M/X. A module M is said to satisfy the finite internal exchange propertyif, for any direct summand X of M and any finite direct sum decomposition M=Lni=1Mi, there exists a direct summand M′i of Mi (i= 1,2, . . . , n) such that M=X⊕(Lni=1M′i). In this paper, we first give characterizations forthe square of a hollow and uniform module to be lifting (extending). In addition, we solve negatively the question "Does any lifting module satisfy the finite internal exchange property?" as an application of this result.