Central Drazin Inverses And One-sided Central Drazin Inverses

We introduce and study a subclass of Drazin inverses,which is called to be central Drazin inverses.Compared with traditional generalized inverses,central Drazin inverses have some better properties.For example,it is well-known that the reverse order law(ab)-1=b-1a-1 for Iost generalized inverses is not true,but if a,b are central Drazin invertible and i(a),i(b)=1,then(ab)D=bDaD holds.We also study the one-sided version of central Drazin inverses which serves as a natural generation of central Drazin inverses and one-sided inverses.Some applications of one-sided central Drazin inverses in ring theory are given.In Chapter 2,we define and study central Drazin inverses.We first study the condition under which a regular element has a unique reflexive inverse.We show that a?R has a unique reflexive inverse if and only if a is group invertible and a#a?C(R).Then we study the intertwining properties of Drazin inverses.Let d?R be Drazin invertible.It is proved that d is central Drazin invertible if and only if da=bd implies adD=dDb for every a,b?R.What's more,we study Cline's formula for central Drazin inverses.A new equivalent condition for a ring to be Dedekind-finite is obtained.Finally,we consider decompositions of central Drazin invertible elements under the addition and the multiplication of a ring.In Chapter 3,we introduce and study a natural generation of central Drazin inverses and one-sided inverses,that is,one-sided central Drazin inverses.Firstly,we study the relationship between central Drazin inverses and one-sided central Drazin inverses.It is proved that an element a is central Drazin invertible if and only if a is both left and right central Drazin invertible.Moreover,we study the relationship between one-sided central Drazin inverses and one-sided inverses.We use one-sided invertible elements to characterize one-sided central Drazin invertible elements.Secondly,we study the counting problem of one-sided central Drazin inverses of one-sided central Drazin invertible elements,and generalize Kaplansky's results on one-sided inverses.Finally,we give some applications of one-sided central Drazin inverses in ring theory.Some new characterizations of Dedekind-finite rings are obtained.