Copolyform \(\Sigma\)-lifting modules. (Q1762785)

Let \(M\) be a right \(R\)-module (\(R\) is a unital associative ring). If \(A\leq B\leq M\) then \(B\) is a `coessential extension' of \(A\) in \(M\) if \(B/A\) is a small submodule of \(M/A\). Under the same hypothesis \(B\) is a `corational extension' of \(A\) in \(M\) if \(\Hom(M/A,B/X)=0\) for all \(A\leq X\leq B\). A submodule \(A\leq M\) is `coclosed' in \(M\) if \(A\) has no proper coessential submodules in \(M\). A module \(M\) is called `copolyform' if every coessential extension is a rational extension. The main result of the paper is Theorem 3.9. In this theorem the authors prove that if \(M\) is a copolyform \(\Sigma\)-lifting module then \(M\) is a direct sum of local self-projective modules whose endomorphism rings are division rings. Moreover, if \(M\) is finitely generated then \(\text{End}(M)\) is a left and right serial, right Artinian, left and right hereditary ring, \(M\) has a projective cover \(P\) in \(\sigma[M]\), and the endomorphism ring of \(P\) is a semisimple ring.