Triangular Decomposition For Matrices In The 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 Yiqiang about a class of formal matrix rings,which introduces a number of properties in.formal matrix ringMn(R;S)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 five chapters.The first chapter,we introduce some basic concepts and symbols,as well as the main background related to the n×n formal matrix ring.And do some promotion of my own understanding.The second chapter mainly introduces the Bass stable rank 1 and proves that when the commutative ring has Bass stable rank 1 property,the reversible matrix in Mn(R;s)can be decomposed into the product of three triangular matrices.In the third chapter,based on the conclusion of the second chapter,we introduce the triangular decomposition of the general matrices in the formal matrix ring over the domain F.Firstly,we prove that the matrix in Mn(F;S)can be expressed as a product of an inverse matrix and a lower triangular matrix.The most important theorem of this chapter is that all the matrices in Mn(F;s)can be decomposed into the product of three triangular matrices.Finally,the triangular decomposition formulas of all matrices in M2(F;s)and M3(F;s)are calculated.In the fourth chapter,on the basis of the previous two chapters,we will study the triangular decomposition for the matrices in Mn(R;s)where R is a commutative rings,and then define the s-Hermite ring,the s-Hermite triangular form,the s-PL property,and prove The equivalence between the three.Finally,we obtain and prove the equivalence condition that the matrices in Mn(R;s)can be decomposed into the product of three triangular matrices.In the last chapter,we study the triangular decomposition of the matrices in M3(R;s)when R is a weakly stable ring.Firstly,we give the definition and equivalent characterization of weakly stable rings and introduce the properties of weakly stable rings The Finally,it is proved that when R is a weakly stable ring,the matrices in M2(R;s)can be decomposed into the product of three triangular matrices if and only if R is a right s-Hermite ring.