It has been shown by
Nǎstǎsescu and Popescu that every nonzero (left, unital) module over a ring R has
a simple submodule if and only if the Jacobson radical J of R is right T-nilpotent
and every nonzero R∕J-module has a simple submodule. The work presented here
arose largely from an attempt to find a general framework for results like
this.
In §2 it is shown that if R has a right T-nilpotent ideal I, then a bijection from
the torsion classes of R∕I-modules to those of R-Modules can be obtained by
associating with each 𝒯 ⊆Mod(R∕I) the lower radical class it defines as a class of
R-modules. §3 contains applications involving the lifting of torsion properties and in
§4 it is shown that if R has a right T-nilpotent ideal I such that R∕I is
the direct sum of its torsion and divisible ideals, then R has this property
also.
