Weak Krull-Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension. (Q706019)

A right (or left) module \(U\) over a ring \(R\) is uniserial if its submodules form a chain under inclusion. A module \(M\) is called serial if it is a direct sum of uniserial modules. The main result of this paper asserts that direct summands of serial modules of finite Goldie dimension are also serial. This gives a positive answer to a question studied by \textit{A. Facchini} and the reviewer [J. Pure Appl. Algebra 133, No. 1-2, 93-106 (1998; Zbl 0936.16005)], where it is shown that any direct summand of a finite direct sum of copies of a uniserial module \(U\) is also a direct sum of copies of \(U\). The author also generalizes Facchini's weak Krull-Schmidt theorem on serial modules of finite Goldie dimension for a larger class of modules whose endomorphism rings have finitely many maximal right ideals and every maximal right ideal is two-sided.