On the composite of irreducible morphisms in almost sectional paths. (Q2456399)

Let \(A\) be an Artin algebra, and \(\text{mod\,}A\) the category of all finitely generated left \(R\)-modules. For indecomposable modules \(X\) and \(Y\) in \(\text{mod\,}A\), a morphism \(f\colon X\to Y\) is called irreducible if \(f\) does not split, and whenever \(f=gh\), then either \(h\) is a split monomorphism or \(g\) is a split epimorphism. It is well-known that an irreducible morphism \(f\colon X\to Y\) belongs to the radical \(\mathfrak R(X,Y)\), but not in its square \(\mathfrak R^2(X,Y)\). For a non-zero composite \(g=f_n\cdots f_1\colon X_0\to X_n\) of \(n\) irreducible morphisms \(f_i\), it is not always true that \(g\in\mathfrak R^n(X_0,X_n)\setminus\mathfrak R^{n+1}(X_0,X_n)\). In a recent paper [J. Algebra 312, No. 2, 650-667 (2007; Zbl 1151.16018)], the authors considered the question of when the composite of two irreducible morphisms is non-zero and belongs to \(\mathfrak R^3(\text{mod\,}A)\). In the paper under review, the authors study the more general situation of when the composite of \(n\) irreducible morphisms in almost sectional paths is non-zero and belongs to \(\mathfrak R^{n+1}(\text{mod\,}A)\).