Study On Quasipolar Element In The Formal Matrix Ring

In this paper,we study the properties of the quasipolar element on the formal matrix ring.With the basic properties of the formal matrix ring as the research tool,the quasipolar element on the formal matrix ring is studied and the conclusion is drawn.First of all,we mainly introduce the basic definitions and symbols of the formal matrix rings involved in this paper,as well as the main background knowledge of the formal matrix ring,and give my own understanding promotion.Secondly,the weak nil-quasipolar elements on the formal matrix ring are explored,in the chapter 2,the equivalence between the weak nil-quasipolar elements and the characteristic equations on the2-order formal matrix ringsM₂?R;s?on the commutative local ring R are given.Characterization of weak nil-quasipolar elements on subrings of n-order formal matrix rings M_n?R;s?on the noncommutative local rings R.And the equivalent characterization of weak nil-quasipolar elements on₂T?R;s?over the commutative domain R is given.Thirdly,on the basis of the second chapter,the weak J-quasipolar elements on the formal matrix ring are further explored,and the equivalent characterization of the weak J-quasipolar elements of the2-order formal matrix ring₂M?R[[x]];s?on the commutative power series ringR[[x]]is given.Furthermore,the weak J-quasipolar elements on a special 3-order formal matrix ring are explored,and the characterization of weak J-quasipolar elements on the sub-rings of 3-order formal matrix ring over the noncommutative local rings is given.Finally,the quasipolar elements on the n-order formal matrix ring M_n?F;s?over the algebraic closed field F are explored,and it is proved that the elements on the n-order formal matrix ring M_n?F;s?are quasipolar elements.In particular,the elements on the n-order formal matrix on the complex field are quasipolar elements.