Some New Characterizations Of Normal Elements In Rings

Combining the generalized inverses theory to study the properties of elements in the ring is a new research method,which has attracted many scholars to carry out a series of in-depth research from various angles.In recent years,many research on EP elements,normal ele-ments and symmetric elements in the ring has been obtained.The remarkable results have led to many new concepts,and the research on these new concepts has become more and more mature.However,there are still many problems to be solved.This paper mainly constructs some specific equations in the context of associative rings with identities,and studies the new properties of normal elements by studying their solutions.The normal element is related to other elements in the ring,and the equivalent condition of the normal element is given.The whole paper is made up of three chapters.In Chapter 1,some basic knowledge points and related theorems involved in this paper are introduced.Chapter 2,mainly studies the necessary and sufficient conditions for the elements in the ring to be normal elements under different conditions.This chapter is mainly divided into two sections.First,the properties of normal elements are studied in combination with the theory of modules.The necessary and sufficient conditions for the elements in the ring to be normal elements are given by the definition,and properties of the small submodules and idempotent ideals.Secondly,the partial equidistant elements on the ring,the strong partial order equidistant elements and the Bott-Duffin invertible element are associated with the normal elements to study the equivalent conditions of the normal elements.In Chapter 3,we obtain some new properties of the normal element,and give the equiv-alence between the normal element and the solution of some specific equations.