Maximal zero product subrings and inner ideals of simple rings
  Let Q be a (non-unital) simple ring. A nonempty subset S of Q is said to have zero product if S^2=0. We classify all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra. 
Alexander Baranov and Antonio Fern\'andez L\'opez
math.RA