Given A an artin algebra we 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. In particular, when A is a finite dimensional algebra over an algebraically closed field we prove that if 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.

中文翻译：

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关于自由基的第五幂的四个不可约态态的组成。

鉴于甲Artin代数我们当四个束缚态射不可分解之间的非零组合物研究甲-模块属于自由基其模块类别的第五功率。特别地，当A是代数闭合域上的有限维代数时，我们证明如果不可分解A-模之间的四个不可约态态的成分属于其模块类别的根的第五次幂，则之间的四个不可约-态的成分相同的不可分解的A模块也是如此。