  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.
