Vol. 40, No. 1, 1972

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Abelian groups which admit only nilpotent multiplications

William Jennings Wickless

Vol. 40 (1972), No. 1, 251–259

Let G+be an abelian group. A ring R with additive group R+isomorphic to G+is a ring on G. G is nil (radical) if and only if R2 = (0)( R is nilpotent) for all rings R on G. It is shown that G is a mixed radical group if and only if T is divisible and G∕T is radical, where T is the maximal torsion subgroup of G. Thus, the study of radical groups is reduced to the torsion free case. A torsion free group G is of field type if and only if there exists a ring R on G such that Q R is a field. It is shown that a torsion free group of finite rank is radical if and only if it has no strongly indecomposable component of field type. It follows that finite direct sums of finite rank radical groups are radical. If G is torsion free an element x G is of nil type if and only if the height vector h(x) = mxis such that 0 < mi < for infinitely many i. Multiplications on torsion free groups all of whose nonzero elements are of nil type are discussed under the assumption of three chain conditions on the partially ordered set of types. Two special classes of rank two torsion free radical groups are characterized. An example is given of a torsion free radical group homogeneous of non-nil type, and a simple condition is given for such a homogeneous group to be nonradical.

Mathematical Subject Classification 2000
Primary: 20K15
Received: 21 December 1970
Revised: 19 July 1971
Published: 1 January 1972
William Jennings Wickless