On left self distributive rings (Q1917165)

A ring is said to be left distributive if it satisfies the identity \(xyz=xyxz\). The following problem was recorded by \textit{G. F. Birkenmeier, H. Heatherly} and \textit{T. Kepka} [ibid. 60, No. 1/2, 107-114 (1992; Zbl 0782.16018)]: Is it true that every left self distributive ring is a direct sum of its nilradical and a left ideal which is a Boolean ring. This is true for left self distributive rings satisfying various additional conditions [see loc. cit., Theorem 3.7 and Corollaries 3.4, 3.8]. We shall show that in general the answer is negative. Theorem 1. There exists a left self distributive ring \(R\) such that the radical of \(R\) is not a semidirect summand.