Let $R$ be a ring such that every zero divisor is 

nilpotent. We call such a ring a \d. We give 

the structure of periodic $D$-rings, 

weakly periodic $D$-rings, Artinian \ds, 

semiperfect \ds, von Neumann regular \ds, 

{\ds} satisfying certain polynomial identities, 

and semiprime \ds. We also include some 

indecomposability results.