Vol. 30, No. 2, 1969

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A radical coinciding with the lower radical in associative and alternative rings

William Grenfell Leavitt and Yu-Lee Lee

Vol. 30 (1969), No. 2, 459–462

In a recent paper by the second author a construction was given which was shown to coincide with the lower radical in all associative rings. In the present paper this construction is considered in various classes of not necessarily associative rings. It is shown that while the construction still defines a radical, it will in general properly contain the lower radical. More precisely, it is shown that the radical constructed coincides with the lower radical if the semisimple class of the lower radical is hereditary (or, equivalently, if the radical of a ring always contains the radicals of all its ideals).

Mathematical Subject Classification
Primary: 17.50
Secondary: 16.00
Received: 23 May 1968
Published: 1 August 1969
William Grenfell Leavitt
Yu-Lee Lee