Direct sums of semi-projective modules. (Q2893053)

In this quite interesting paper, the authors study when the direct sum of semi-projectve modules is semi-projective. Their main interest is: if \(R\) is a right Ore domain with right quotient division ring \(Q\neq R\) and if \(X\) is a projective (right) \(R\)-module, when is \(Q\oplus X\) semi-projective?NEWLINENEWLINE They start, in Section 2, with basic properties of semi-projective modules. They prove that every nonsingular, extending (in particular, injective or semisimple) module is semi-projective. If an \(R\)-module \(M\), over a Dedekind domain \(R\), is a direct sum of cyclic submodules (in particular, if \(M\) is finitely generated), then \(M\) is semi-projective if and only if it is quasi-projective. If \(R\) is a prime, right Goldie ring and if \(M\) is an \(R\)-module which is a direct sum of a torsion-free, divisible submodule \(X\) and a torsion, semisimple submodule \(Y\) (here no need to assume \(Y\) to be torsion) then \(M\) is semi-projective. A divisible \(R\)-module over a prime PI ring \(R\) is semi-projective if and only if it is nonsingular.NEWLINENEWLINE In Section 3, they study modules over a right Ore domain \(R\) with right quotient division ring \(Q\neq R\). They prove that every finitely generated submodule of \(Q\) is semi-projective. Further if \(X\) is a projective right \(R\)-module with minimal generating set of cardinality strictly less than that of \(Q_R\) then \(Q\oplus X\) is semi-projective. Further if \(X\) is finitely generated then \(Q\oplus X\) is semi-projective if either \(R\) is right Noetherian or is left Ore.NEWLINENEWLINE Finally, in Section 4, they study modules over a principal ideal domain \(R\) with quotient field \(Q\neq R\). They prove that if \(X\) is a proper submodule of \(Q\) containing \(R\) then \(X\oplus R\) is semi-projective if and only if \(X\) is finitely generated.