Stable Rank And Exchange Of Associative Rings

This dissertation, consisting of four chapters, is concerned with stable rank, diagonalization of matrices over rings and exchange ring. The dissertation falls into four parts:The first chapter presents the background of our work.The second chapter focuses on unit ideal-stable rank, including unit〈I〉-stable rank and unit (I)-stable rank. Let I be an ideal of a ring R, then R satisfies unit〈I〉-stable rank if and only if R satisfies unit (I)-stable rank and I has stable rank one; R satisfies unit〈I〉-stable rank if and only if TM_n(R) satisfies unit〈TM_n(I)〉-stable rank; R satisfies unit〈I〉-stable rank if and only if Whenever aR + bR = dR with a∈1 + I,b∈I and d∈1 +I, there exist u,v∈U(R) such that au+bv=d if and only if Whenever Ra+Rb = Rd with a∈1 + I,b∈I and d∈1 + I, there exist u,v∈U(R) such that ua + vb = d. Let I be a regular ideal of R, R satisfies unit〈I〉-stable rank, if aR + bR = dR with a,b,d∈I, there exist u,v∈U(R) such that au + bv = d. The final section focuses on the exchange ideal satisfying (?)-comparability and prove that the module extension I (?) M of I by M is an exchange ideal satisfying (?)-comparability if and only if so is I , the ideal of Morita context (A,B,N,M,θ,ψ) with zero pairs is an exchange ideal satisfying (?)-comparability if and only if so are A and B, the ideal of power series I[[x₁,x₂,…,x_n]] is an exchange ideal satisfying (?)-comparability if and only if I is an exchange ideal satisfying (?)-comparability.The third chapter studies the diagonalization of matrices over ideals. Let R be a commutative ring, I an ideal, A is an idempotent matrix over I, A is diagonalable under a similarity transformation if and only if A has an I-characteristic vecter; if A is diagonalable under a equivalent transformation if and only if A is diagonalable under a similarity transformation ( where the invertible matrix is written as a sum of an identity matrix and a matrix over I).The fourth chapter investigates Morita contexts over regular QB-rings and exchange ideals satisfying (?)-comparability. Let A, B be ideal of R and S, respectively, _BM_A, _AN_b are bi-modules, we prove that each element in (A,B,M,N,θ,φ) can be written as a sum of an idempotent and a pseudo-invertible element if A and B are regular QB-ideal or regular ring satisfying general comparability. We demonstrate that the elements the ideal of Morita contexts over two regular QB-ideals can be written as above.In the final chapter, we study some properties of 2-power stably free rank. For the 2-power stably free modules, we give some properties of 2-power stably resolutions and Euler characteristic over IBN rings.