Similar Classification Of Idempotent Matrices Of Formal Matrix Rings Over A Local Ring And Its Applications

Formal matrix rings as a generalization of the matrix rings,played an important role in ring theory and module theory,and they were studied by many scholars.It is well known that idempotent elements in a ring are directly related to the direct sum decomposition of this ring.And idempotent matrices are a class of matrices with good properties in matrix rings.Idempotent matrices not only play an important role in the decomposition of diagonal matrices,but its idempotency corresponds to the projection operator that provides a tool for the study of the projection of vector space to its subspace direction.The idempotent matrices of a formal matix ring are studied that can make the structure of this formal matrix ring clearer.Therefore,it is of great significance to study idempotent matrices of a formal matrix ring.In 2009,Wang and Liu et al.in[34]proved that if A is a non-trivial idempotent matrix of matrix ring Mn(R)over a commutative local ring R,then A-diag{Er,0},where 1 ≤r ≤n.In this thesis,we proved that if A is an idempotent matrix of formal matrix ring Mn(R;∑n)over a local ring R,then A～diag{d11,d22,…,dnn},where dii=0 or 1,i=1,2,···,n.In addition,we discussed whether the formal matrix ring over a field is a strongly clean ring.And we completely characterize whether every matrix of the formal matrix ring over a field is a sum of three idempotent matrices.Chapter 0 introduced the research background of this thesis and its significance.Chapter 1 gave some relative basic concepts.Chapter 2 mainly studied the similar classification of idempotent matrices of formal matrix rings M n(R;∑n)over a local ring R.And we proved that if A is an idempotent matrix of formal matrix ring Mn(R;∑n)over a local ring R,then A-diag{d11,d22,···,dnn},where dii=0 or 1,i=1,2,…,n.Chapter 3 mainly discussed whether the formal matrix ring over a field is a strongly clean ring.And we proved that Mn(F;0)is a strongly clean ring,where F is a fieldChapter 4 mainly studied whether every matrix of the formal matrix ring over a field is a sum of three idempotent matrices.And we proved that every element of formal matrix ring Mn(F;0)over a field F is a sum of three idempotent elements if and only if F(?)Z2,1≤n≤4 or F(?)Z3,1≤n ≤2.