The Moore-Penrose inverse in rings with involution
  Let $R$ be a unital ring with involution.In this paper, several new necessary and sufficient conditions for the existence of the Moore-Penrose inverse of an element in a ring $R$ are given.In addition, the formulae of the Moore-Penrose inverse of an element in a ring are presented. 
Sanzhang Xu, Jianlong Chen
math.RA