On lifting LE-modules. (Q1862681)

A module \(M\) is said to be lifting if, for any submodule \(N\) of \(M\), there is a decomposition \(M=M_1\oplus M_2\) for which \(M_1\subseteq N\) and \(N\cap M_2\) is small in \(M\). The authors of the paper under review call a module \(M\) an LE-module if it has a decomposition \(M=\bigoplus_{i\in I}M_i\) where each \(M_i\) has a local endomorphism ring. They also say that a lifting LE-module \(M\) satisfies (*) if any direct sum of \(M\) and a semisimple module is also a lifting module. Their first main result is that any lifting module \(M\) decomposes as \(M=M_1\oplus M_2\oplus M_3\) where \(M_1\) is semisimple, \(M_2\) is a lifting module in which the radical is both small and essential, and \(M_3\) is a lifting module with no (proper) maximal submodules; if moreover the radical of \(M\) satisfies ACC or DCC on direct summands then \(M_2\) and \(M_3\) are both finite direct sums of hollow modules. Then, following a detailed investigation of the lifting property in conjunction with local semi-T-nilpotency and projectivity conditions, they show that the following conditions are equivalent for a ring \(R\): (1) \(R\) is right perfect and every lifting right LE-module satisfies (*); (2) \(R\) is semiperfect and every finitely generated lifting right LE-module satisfies (*); (3) \(R\) is semiperfect and any direct sum of a simple and a projective local right \(R\)-module is lifting; (4) \(R\) is semilocal and the square of its Jacobson radical is zero; (5) the left-handed versions of (1), (2), or (3) hold.