Rings with left self distributive multiplication (Q1207357)

An associative ring is left self distributive (LD-ring, for short) if it satisfies the identity \(xyz = xyxz\). Analogously right self distributive rings (RD-rings) are introduced. Rings which are both LD and RD-rings were already classified by M. Petrich. The present paper shows that LD- rings form a much more extensive variety. If \(N\) denotes the set of all nilpotent elements of an LD-ring \(R\), then \(N\) is an ideal, \(N^ 3 = 0\), and \(R/N\) is Boolean. Moreover if \(R/N\) contains a unity then \(R = A\oplus N\) (direct sum of left ideals) with \(A\) a Boolean ring with unity. Further it is proved that any LD-ring is left permutable, i.e. satisfies \(xyz = yxz\) identically, and hence medial. Finally a complete description of subdirectly irreducible LD-rings is given: A subdirectly irreducible LD-ring is either nilpotent of index at most three, \(Z_ 2\), or a certain four element ring.