Some Properties Of Semirings And Semimodules

Semirings as a common algebraic system,which have important applications in topology,analysis,computer science and so on.Semimodules are an effective method to study the structure of semirings,as a representation of semirings.Symmetries have not been extended to semirings and semimodules,so this paper mainly studies the symmetries of semirings and semimodules.We generalize the corresponding results of the symmetries of rings and modules Main contents are discussed as followsChapter 1 introduces the research background and current situation of the symmetries of semirings and semimodules,the reducible package and annihilator,and the main results of this paperChapter 2 introduces some elementary definitions and notations that will be used.Chapter 3 discusses the symmetries of semimodules and semirings.In this chapter,the several classes concepts of symmetric semirings,symmetric semimodules,generalized weakly symmetric semirings,and central symmetric semirings are defined.Their basic properties and the relationship between the semirings classes such as symmetric semirings and reduced semirings are discussed.The equivalence relationship between Abel semimodules and symmetric semimodules are given,basing on p·p semimodules.The triangular matrix semirings for the generalized weakly symmetric semirings are generalized weakly symmetric.Chapter 4 is concerned with the ideal symmetries of semirings and semimodules.In this chapter,the ideal symmetric semirings and ideal symmetric semimodules are defined.The basic properties of ideal symmetric semirings and semimodules are discussed,and give the necessary and sufficient conditions of symmetric and ideal symmetric.We define the trivial extension of semirings R through semimodules M.Prove the trivial extension of a semiprime semirings on a regular left R semimodules are ideal symmetric,and the matrix semirings of order n on ideal symmetric semirings R are ideal symmetricChapter 5 discusses the reducible package and annihilator.The reducible package is defined.The basic properties of reducible package and order pair annihilator are discussed.The equivalent characterization of p-injectivity and generator are given,as an application.It is proved that if R/?I is ideal symmetric,then R is ideal symmetric,where I is a reducible ideal without nonzero nilpotent elements of R.