Strong lifting splits. (Q534011)

In this paper the concept of an enabling ideal is introduced. An ideal \(I\) of \(R\) is called an enabling ideal of \(R\) if \(a-e\in I\), \(a\in R\), \(e^2=e\in R\), then \(a-f\in I\) for some \(f^2=f\in Ra\). The authors show that an ideal \(I\) of \(R\) is strongly lifting if and only if \(I\) is lifting and enabling. These ideals are studied and their properties are described. It is shown that a left duo ring is an exchange ring if and only if every ideal is enabling, that Zhou's \(\delta\)-ideal is always enabling, and that the right singular ideal is enabling if and only if it is contained in the Jacobson radical. The notion of a weakly enabling left ideal is defined. They characterize a weakly enabling ideal as a left ideal (right ideal) by left (right) partial summand of the ring. It is shown that a ring is an exchange ring if and only if every left ideal is weakly enabling. They show that if \(R\) is a left SSP ring, then an ideal \(I\) of \(R\) is a weakly enabling ideal of \(R\) if and only if \(I\) is enabling. Two related conditions, interesting in themselves, are investigated: the first gives a new characterization of \(\delta\)-small left ideals and the second characterizes weakly enabling left ideals. As an application (which motivated the authors), let \(M\) be an \(I\)-semiregular left module where \(I\) is an enabling ideal. It is shown that \(m\in M\) is \(I\)-semiregular if and only if \(m-q\in IM\) for some regular element \(q\) of \(M\) and, as a consequence, that if \(M\) is countably generated and \(IM\) is \(\delta\)-small in \(M\), then \(M\cong\bigoplus^\infty_{i=1}Re_i\) where \(e^2_i=e_i\in R\) for each \(i\).