The 𝒯 -divisible R-modules are
defined in terms of a hereditary torsion theory of modules over an associative ring R
with identity element. In the special case where 𝒯 is the usual torsion class of
modules over a commutative integral domain, the class of 𝒯 -divisible modules is
precisely the class of divisible modules M such that every nonzero homomorphic
image of M has a nonzero h-divisible submodule. In general, if 𝒯 is a stable
hereditary torsion class, the class of Z-divisible modules satisfies many of the
traditional properties of divisible modules over a commutative integral domain. This
is especially true when F is Goldie’s torsion class 𝒢. For suitable 𝒯 the splitting of all
𝒯 -divisible modules is equivalent to h.d. Q𝒯≦ 1, where Q𝒯 is the ring of quotients
naturally associated with 𝒯 Generalizations of Dedekind domains are studied in
terms of 𝒯 -divisibility.
