On the square of a uniserial module (Logic, Language, Algebraic system and Related Areas in Computer Science)

Abstract

A module M is said to be lifting if, for any submodule 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 property if, for any direct summand X of M and any finite direct sum decomposition M = ⊕^ni=1 Mi, there exists M´i ⊆ Mi (i = 1, 2, ... , n) such that M = X⊕(⊕^ni=1 M´i). In this paper, we solve negatively the open problem "does any lifting module satisfy the finite internal exchange property?" by considering the square of a certain lifting module.