Dear Fawaz,

1. Prime gamma ring A gamma ring R is called prime if for any elements a,b in R, the condition aΓRΓb={0} implies either a=0 or b=0, meaning the product through the gamma operation is zero only when one element is zero.

2. Semiprime gamma ring A gamma ring R is semiprime if it has no non-zero nilpotent gamma ideals equivalently if aΓRΓa=0 for some a in R then a=0.

3 Completely prime gammma ring A gamma ring R is completely prime if for any a,b in R, aГb=0 implies either a=0 or b=0, which is a stronger condition than prime requiring the direct gamma product to be zero only when one element is zero.

Hope it helps :-)