Moore-Penrose inverse and normal elements in rings

Abstract

Let R be a ring with involution*, then a € R is a Moore-Penrose invertible element if there is bЄR such that aba = a, bab = b, (ab)* = ab and (ba)* = ba. b is called Moore-Penrose inverse of a, denoted by a1 (if it exists). In this thesis, we study Moore-Penrose inverses and normal ele- ments in ring with involution and give the neccessary and sufficient conditions for the existence of the Moore-Penrose inverse of an element in ring with involution. Furthermore, we also investigate the existence of the Moore-Penrose inverse for the product 123. En where T1, T2,..., In are Moore-Penrose invertible.