The Structure And Equivalent Characterizations Of A(?)(?)~f I Ring

As an important subject of algabra,ring theory is the basis of many fields such as algebraic geometry and algebraic number theory.The amalgamation of A with B along I with respect to f,which is denoted by A(?)(?)~f I,is a new extension method.We can enrich ring theory by studying it theoretically.In recent years,scholars continue to optimize the definition of A(?)(?)~f I.And some equivalent conditions between commutative A(?)(?)~f I ring and other rings are studied.In this paper,the nonlcommutative definition and basic constructions of A(?)(?)~f I ring are given.And then,the relationship between these rings are also given.Specific works include the following:In the first chapter,the ralated background knowledge of A(?)(?)~f I ring,some classes of clean rings,regular rings,Armendariz rings and McCoy rings are given.In the second chapter,under the non-commutative conditions,the definition of A(?)(?)~f I ring,the basic constructions of idempotent elements,invertible elements and the Jacobson radical are given.In the third chapter,the relationship between non-commutative A(?)(?)~f I ring and clean ring are studied.Firstly,some equivalent conditions between non-commutative A(?)(?)~f I ring and(unique)clean rings,2-clean rings are proved.Secondly,the nil-clean property of commutative A(?)(?)~f I I ring is given.Finally,the equivalent conditions between regular ring and non-commutative A(?)(?)~f I ring are studied.In the fourth chapter,the new properties of A(?)(?)~f I ring are proved.Some equivalent conditions between A(?)(?)~f I ring and(weakly)Armendariz ring are studied.Finally,some McCoy properties of A(?)(?)~f I ring are extended.