Study On Properties Of Clean Element In 2×2 Formal Matrix Ring

In ring theory,different matrices have different functions,which the formal matrix ring occupies a very important position.In the paper of Tang Gaohua and Zhou Yijiang about a class of formal matrix rings,which introduces a number of properties in formal matrix ring M_n(7)R;s(8)that defined by a central element s in R.This article is inspired by this idea,extending some properties of the formal matrix ring through central element s and draw some relevant conclusions.This thesis mainly consists of four chapters.The first chapter,we introduce some basic concepts and symbols,as well as the main background related to the2?2formal matrix ring.And do some promotion of my own understanding.The next chapter,we mainly introduce the Jacobson radical and the equivalent forms of strongly clean element on the2?2formal matrix ring over local rings.Comprehensive some drawing of strongly clean element and strongly clean ring.The third chapter,on the basis of our second chapter,we do further studied about the strongly clean properties of the2?2formal matrix ring over the exchanged local ring.At this point we can use the s-determinant on the formal matrix ring defined in Chapter one,then the equivalent form of strongly clean element and strongly clean ring over the exchanged local ring is obtained.The last chapter,based on the theory of the second and third chapters,the formal matrix ring based on special rings is studied.We discussed in turn about residue class ring defined by a prime number p and the localization of integer ring Z at a prime ideal which generated by a prime number p.We can further see the equivalent forms of strongly clean element on these special rings.Besides,we prove that the formal matrix ring on the complex number field and real number field are both strongly clean rings.