We study direct-sum decompositions of torsion-free, finitely generated modules over a (commutative) Bass ring R through the factorization theory of the corresponding monoid T(R). Results of Levy–Wiegand and Levy–Odenthal together with a study of the local case yield an explicit description of T(R). The monoid is typically neither factorial nor cancellative. Nevertheless, we construct a transfer homomorphism to a monoid of graph agglomerations—a natural class of monoids serving as combinatorial models for the factorization theory of T(R). As a consequence, the monoid T(R) is transfer Krull of finite type and several finiteness results on arithmetical invariants apply. We also establish results on the elasticity of T(R) and characterize when T(R) is half-factorial. (Factoriality, that is, torsion-free Krull–Remak–Schmidt–Azumaya, is characterized by a theorem of Levy–Odenthal.) The monoids of graph agglomerations introduced here are also of independent interest.

我们通过相应的类半体T（R）的因式分解理论研究（交换）Bass环R上无扭转的，有限生成的模块的直接和分解。Levy–Wiegand和Levy–Odenthal的结果以及对本地案例的研究得出了对T（R）的明确描述。该类半体通常既不是阶乘也不是取消式的。尽管如此，我们还是将转移同态构造为图集的一个单面体-一个自然类的单面体，用作T（R）分解理论的组合模型。结果，the半群T（R）是有限类型的传递Krull，并且对算术不变量应用了几个有限性结果。我们还建立了关于T（R）弹性的结果，并描述了当T（R）为半因子时的特征。（功能性，即无扭转的Krull–Remak–Schmidt–Azumaya的特征是Levy–Odenthal定理。）这里介绍的图集的等分线也具有独立的意义。