Vol. 53, No. 1, 1974

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Some aspects of T-nilpotence

Barry J. Gardner

Vol. 53 (1974), No. 1, 117–130

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.

Mathematical Subject Classification
Primary: 16A22
Received: 1 June 1973
Published: 1 July 1974
Barry J. Gardner