A number of questions
involving T-nilpotence are studied. §1 contains characterizations of left and two-sided
T-nilpotent rings in terms of (transfinite) annihilator series and a list of ring
constructions which preserve T-nilpotence. In §2 the radical theory of T-nilpotence is
investigated. It is shown that a left T-nilpotent ring belongs to a radical
(resp. semisimple) class precisely when the zeroring on its additive group does so,
and that there are no interesting radical classes which consist entirely of left
T-nilpotent rings. §3 is devoted to an examination of the effect which chain
conditions on the type set of a suitably restricted torsion-free abelian group G have
on the kinds of ring multiplication which G admits. Some conditions are given
which are sufficient to ensure that every multiplication on G is (two.sided)
T-nilpotent. A result from §2 is used to show that certain homogeneous
groups do not admit nontrivial nilpotent multiplications. In the final brief
section an example is used to show that whereas two-sided T-nilpotent rings
satisfy the idealizer condition, the same need not be true of a left T-nilpotent
ring.
