Some Characterization Of Strongly Mary Inverse

An element a in semigroup S is said to be Strongly invertible along d if there exists y∈S,such that ayd=d=dya,y∈Sd(?)dS.which is denoted by as‖d.a is Strongly invertible along d is the generalization of a is invertible along d.In the second chapter of this Master’s thesis,we study some properties of a is invertible along d,and we prove the following results:(1)a‖d exists if and only if d∈dadSad ∩ daSdad;(2)a‖d exists if and only if the equation d=daxad has at least one solution in Sd∩dS;(3)a‖d exists if and only if the equation d=daxdad has at least one solution in comml(d).In the third chapter,we investigate some properties of a is Strongly invertible along d as follows:(4)as‖d exiests if and only if d∈adSd∩dSda∩Sdad∩dadS;(5)as‖d exiests if and only if d∈adSdn∩dnSda,for any n≥2.In the fourth chapter,we discuss the applications of a is invertible along d to rings as follows:(6)Ris an Abel ring if and only if for any e,g ∈ E(R),e‖g exists to imply g=ge;(7)Ris a CN ring if and only if for any a∈N(R)and b,c∈R,c‖(ab-ba)exists.In the fifth chapter,we present the applications of a is Strongly invertible along d to generalized inverse in ring with involution as follows:(8)Let R be a*-ring and a∈R+.Then a∈REP if and only if a+s‖a exists;(9)Let R be a*-ring and a∈R+.Then a∈RPI if and only if a+s‖(aa+aa*)exists and a+-a*∈N(R).