Rickart modules relative to singular submodule and dual Goldie torsion theory (Q2816908)

If \(M\) is any right \(R\)-module, \(f \in S=\mathrm{End}_{R}(M)\) and \(N\) a submodule of \(M\), then \(M\) is said to be N-inverse split if \(f^{-1}(N)\) is a direct summand of \(M\) for every \(f \in S\). In the paper under review, the cases where \(N\) is the singular submodule \(Z(M)\) of \(M\), and where \(N\) is the cosingular submodule \(Z^{*}(M)=\{m \in M \mid mR \text{ is small in } E(M) \}\) (which generates the dual Goldie torsion theory), are studied. The \(Z(M)\)-inverse split modules are characterized as those modules that are a direct sum of \(Z(M)\) and a Rickart module. It is shown that a ring \(R\) is right \(Z(R_{R})\)-inverse split if and only if it is right Rickart, but examples are given to show that this is not true in general with respect to modules. Right nonsingular right hereditary rings are characterized as those modules for which every right projective (or free) module \(M\) is \(Z(M)\)-inverse split.NEWLINENEWLINEThe study of \(Z^{*}(M)\)-inverse split modules starts with several examples of modules that are \(Z^{*}(\cdot)\)-inverse split. These include, amongst others, all small \(R\)-modules and modules over a semisimple ring. An \(R\)-module \(M\) is shown to be \(Z^{*}(M)\)-inverse split if and only if \(M\) is a direct sum of \(Z^{*}(M)\) and a Rickart module. Sufficient conditions for the homomorphic image of a \(Z^{*}(M)\)-inverse split module to be \(Z^{*}(M)\)-inverse split, are found. For several classes of modules it is shown that \(Z^{*}(M)\)-inverse split modules are Rickart, but this is not true in general. It is proved that modules over right hereditary QF-rings are \(Z^{*}(M)\)-inverse split. The paper is concluded with a characterization of the rings \(R\) for which every free (or every projective) \(R\)-module \(M\) is \(Z^{*}(M)\)-inverse split.