Direct summands of serial modules (Q1295651)

The authors investigate the following problem: Is every direct summand of a serial module serial? They have got affirmative answers in some special cases. Every direct summand of a finite direct sum of copies of a uniserial module \(U\) is again a finite direct sum of copies of \(U\). If \(U_1,U_2,\dots,U_n\) are uniserial modules such that for any \(i,j=1,2,\dots,n\), there exists a uniserial module \(W\) such that \([U_i]_m=[W]_m\) and \([U_j]_e=[W]_e\), then every direct summand of \(U_1\oplus U_2\oplus\cdots\oplus U_n\) is serial. Finally, they prove the following: If \(U_1,U_2,\dots,U_n\) are mono (resp. epi)-Krull-Schmidt uniserial modules such that \([U_i]_e \neq[U_j]_e\) (resp. \([U_i]_m\neq[U_j]_m)\) for each \(i\neq j\) in \(\{1,2,\dots,n\}\), then every direct summand of \(U_1\oplus U_2\oplus\cdots\oplus U_n\) is serial.