The endomorphism ring of a nonsingular retractable module (Q2717955)

Let \(R\) be an associative ring with nonzero identity element, let \(M\) be a unital left \(R\)-module, and let \(E=\text{End}_R(M)\) be the endomorphism ring of \(M\). The author investigates the transfer of some properties between \(M\) and \(E\) in case \(M\) is a nonsingular retractable module (\(M\) is said to be retractable if \(\Hom_R(M,U)\neq 0\) for every nonzero submodule \(U\) of \(M\)). Thus, she shows that the ring \(E\) is left quasi-continuous if and only if \(M\) is quasi-continuous, the ring \(E\) is left continuous and regular if and only if \(M\) is continuous, and the left module \(_EE\) has the absolute direct summand property if and only if \(M\) has the absolute direct summand property (this means that whenever we have \(M=A\oplus B\) and \(C\) is a complement of \(A\) in \(M\), then \(M=A\oplus C\)). The author also determines necessary and sufficient conditions on \(M\) which ensure that every left complement in \(E\) is a left annihilator.