The Equation’s Characterations Of SEP Generalized Inverses

Let R be an associative ring with unity and a∈R^#∩R⁺.If a⁺=a^#,then a is called an EP element;if a⁺=a^*,then a is called a PI element;if a⁺=a^*=a^#,then a is called a SEP element.This paper will construct some linear equations of generalized inverse equations,and we will give several new properties of EP elements,PI elements or SEP elements by the solutions of the constructed equations.This paper is organized as follows:In the first chapter,we give the basic definitions and related results and properties.In the second chapter,we first give some necessary and sufficient conditions of EP elements by the properties of EP elements,and then we construct the equations with one variable,and character the existences of EP elements by the solvability of the equations in a given set.The main conclusions are as follow:（1）Let a∈R^#∩R⁺.Then a∈R^EP if and only if for any x∈ρ_a,（axa⁺）+=ax^#a^#;（2）Let a∈R^#∩R⁺.Then a∈R^EP if and only if for any x∈ρ_a,（axa^#）（axa^#）+=a⁺a:（3）Let a∈R^#∩R⁺.Then a∈R^EP if and only if for any x∈ρ_a,（axa⁺+1-aa⁺）^-1=ax^#a^#+1-aa⁺.In the third chapter,we consider the existences of PI elements and SEP elements.The main conclusions are as follow:（4）Assume that the equation xa^*a=xhas solutions,then if x∈τ_a then a∈R^PI;x∈γ_a then a∈R^SEP;（5）Assume that the equation xa⁺（a⁺）*=x has solutions,then if x∈σa then a∈R^PI;if x∈γ_a then a∈R^SEP;（6）Assume that the equation aa^*x=x has solutions,then if x∈η_a then a∈R^PI;if x∈ξ_a then a∈R^SEP.In the fourth chapter,we use the solutions of a^#xa^#=a^#a^*x and its deformation in different sets to character PI elements and SEP elements.Then the equation generalized as a^#xa^#=a^#a^*y,an equation with two variables,and we study the general formula of its solutions.Based on these results,we character SEP elements by the solvability of the equation with two variables.The main conclusions are as follow:（7）Assume that a^#xa^#=a^#a^*x has solutions,if x∈τ_a,then a∈R^PI;if x∈γ_a then a∈R^SEP;（8）a∈R^PI if and only if the general solution of the bivariate equation a^#xa^#=a^#a^*y is given by（?）;（9）a∈R^SEP if and only if there exist x,y∈χ_a² such that yxa^#+a⁺=ya^*x+a^#.