Let \(R\) be an Artin algebra and mod-\(R\) denotes the category of finitely presented right \(R\)-modules. Let rad(mod-\(R\)) be the radical of the category mod-\(R\), consisting of all morphisms \(f\colon X\to Y\) in mod-\(R\) such that the \(R\)-endomorphism \(\text{id}_X-gf\) is invertible for every \(g\) in \(\operatorname{Hom}(Y,X)\). The intersection of all finite powers of rad(mod-\(R\)) is denoted by rad\(^\omega\), and is called the \(\omega\)-radical of mod-\(R\). It is known that any morphism between indecomposable finitely presented \(R\)-modules which lie in different components of the Auslander-Reiten quiver of mod-\(R\) must belong to rad\(^\omega\).
In this paper, the author shows that, for any Artin algebra \(R\), any morphism in rad\(^\omega\) factors through a finite direct sum of indecomposable infinite length pure-injective modules. The proof is based on the concept of fp-idempotent ideals of mod-\(R\), introduced by H. Krause [Mem. Am. Math. Soc. 707 (2001; Zbl 0981.16007)].