Throughout this note all rings are associative, but not necessarily commutat ive or with unity. A ring is left self distributive (an LD-ring) if it satisfies the identity: x y z = xyx z . Similarly one defines a right self distributive ring (an RD-ring). Petrich [7] classified all rings which are both L D and RD-rings as those rings which are the direct sum of a Boolean ring and a nilpotent ring of index at most three. As shown by several examples given herein, this does not hold for all LD-rings. These examples illustrate how rich the variety of LD-rings is. If R is an LD-ring and N is the set of nilpotent elements of R, then N is an ideal, N 3 = 0, and R / N is Boolean. If R / N contains unity, then R = A q~ N as a direct sum of left ideals, with A a Boolean ring with unity. This condition is implied by several others; e.g., R has d.c.c, on ideals or a.c.c, on ideals. Without any finiteness condition we are still able to find an ideal B = A + N, where A is Boolean and is a left ideal, such that B is completely semiprlme and is left and right essential in R. Somewhat surprising from the viewpoint of semigroup theory [5] is the result that every LD-ring is left permutable (satisfies abc= bac identically). Other useful identities are developed. A complete classification of subdirectly irreducible LD-rings is given. Such a ring is either nilpotent of index at most three, Z2, or a certain four element ring.