A study on dual square free modules

Abstract

Let \(M\) be an \(H\)-supplemented coatomic module with FIEP. Then we prove that \(M\) is dual square free if and only if every maximal submodule of \(M\) is fully invariant. Let \(M=\bigoplus_{i\in I} M_i\) be a direct sum, such that \(M\) is coatomic. Then we prove that \(M\) is dual square free if and only if each \(M_i\) is dual square free for all \(i\in I\) and, \(M_i\) and \(\bigoplus_{j\neq i}M_j\) are dual orthogonal. Finally we study the endomorphism rings of dual square free modules. Let \(M\) be a quasi-projective module. If End\(_R(M)\) is right dual square free, then \(M\) is dual square free. In addition, if \(M\) is finitely generated, then End\(_R(M)\) is right dual square free whenever \(M\) is dual square free. We give several examples illustrating our hypotheses.