On the composition of four irreducible morphisms in the fifth power of the radical (Q2048319)

Let \(A\) be an Artin algebra over a fixed commutative Artin ring. We denote by mod\(\,A\) the category of finitely generated right \(A\)-modules and by rad\(_A\) the Jacobson radical of mod\(\,A\). It is well known that a morphism \(f: X\to Y\) between two indecomposable modules \(X\) and \(Y\) in mod\(\,A\) is irreducible if and only if \(f\in\) rad\(_A(X,Y)\setminus\) rad\(_A^2(X,Y)\). Moreover, if \(g: X\to Y\) is a non-zero composition of \(n\geq 2\) irreducible morphisms between indecomposable modules, then it is not always true that \(g\in\) rad\(_A^{n}(X,Y)\setminus\) rad\(_A^{n+1}(X, Y)\) (see [J. London Math. Soc. 45, No. 1, 32--54 (1992; Zbl 0703.16010)] and [J. Algebra 312, No. 2, 650--667 (2007; Zbl 1151.16018)]). In the paper under review, the authors study when the non-zero composition of four irreducible morphisms between indecomposable \(A\)-modules belongs to the fifth power of the radical of its module category (Theorem A). As an application of Theorem A, they prove that if \(A\) is a finite dimensional algebra over an algebraically closed field and the composition of four irreducible morphisms between indecomposable \(A\)-modules belongs to the fifth power of the radical of their module category then any composition of four irreducible morphisms between the same indecomposable \(A\)-modules so is (Theorem B). Finally, let us mention that this is a continuation of previous work [the first author et al., J. Algebra 312, No 2, 650--667 (2007; Zbl 1151.16018); Commun. Algebra 39, No. 2, 555--559 (2011; Zbl 1211.16012)] and others.