1. Introduction Throughout this article, R denotes an associative ring with center Z ( R ) . R is called a prime ring if a R b =0 implies a = 0 or b = 0 and a semi-prime ring if a R a = 0 implies a = 0 for all a , b ∈ R . A ring is said to be two-torsion free if 2 a = 0 (where a ∈ R ) implies a = 0 . In [1], the authors provide a more detailed definition. For all a , b ∈ R , [ a , b ] denotes the Lie product a b − b a and a ∘ b denotes the Jordan product a b + b a .
