Generalized Commuting Properties Of (b,c)-inverses And Its Applications

In this paper,we investigate some commuting properties of a new type of generalized inverses in an arbitrary semigroup S,called?b,c?-inverses which were first introduced by Drazin.We approach these results by first considering the more general case,i.e.,annihilator?b,c?-inverses.The paper is organized as follows.In Chapter 2,we are concerned with an extended version of?b,c?-inverses named anni-hilator?b,c?-inverses.We first show that,contrary to one-sided?b,c?-inverses of elements taken from semigroups,an element in an arbitrary ring could have left?or right?annihilator?b,c?-inverses that are different from its annihilator?b,c?-inverse.We then give some ex-amples to illustrate this point.Several results related to one-sided annihilator?b,c?-inverses and its uniqueness are also presented.In Chapter 3,we consider an intertwining relation,namely,yx₁=x₂y,where y?S,and x_i?S is the?b_i,c_i?-inverse of a_i?S for given b_i,c_i?S,i=1,2.Next,we derive equivalent conditions for each equality below to hold.1)ya₁x₁=a₂x₂y,2)yx₁a₁=x₂a₂y,3)ya₁x₁=x₂a₂y,4)yx₁a1=a₂x₂y.Such equalities can also be reinterpreted by annihilator?b,c?-inverses in rings.We determine their equivalent conditions by only using the relations between a_i,b_i,c_i,i=1,2,and y.It might be emphasized that,to the best of our knowledge,these necessary and sufficient conditions seem not to have been noted before even in the?b,c?-inverse case.In addition,in the one-sided?b,c?-inverse case,we recognize that it can only give us a necessary?or sufficient?condition for intertwining property.In Chapter 4,we study the absorption law and the reverse order law for annihilator?b,c?-inverses,and then for?b,c?-inverses.We show that the absorption law holds,namely,c₁=c₂.Also,we present equivalent conditions for the reverse order law to hold,namely,when the?b₂,c₁?-inverse of a₁a₂coincides with x₂x₁.As corollaries,we propose some general,yet useful results.