Zabavsky B. Diagonal reduction of matrices over rings

In the thesis it is proved that the right (left) Bezout ring of stable rank 1 is a right (left) Hermitian ring. As a consequence, it is shown that any semilocal right (left) Bezout ring is a right (left) Hermitian ring. A criterion is established for a commutative ring to be Hermitian: any commutative Bezout ring is an Hermitian ring if and only if its stable rank is 2. As a consequence, it is shown that any commutative Bezout ring with a compact space of minimal simple ideals is an Hermitian ring. It is proven that any right (left) Bezout ring whose quotient ring by the Jacobson radical is right (left) Hermitian, is also Hermitian. It is proved that any matrix ring over directly finite ring such that every matrix over it is diagonalizable, is directly finite. A diagonal reduction is proved for matrices of certain form over a regular ring of finite stable rank as well as a "weak" diagonal reduction of matrices over arbitrary regular ring. The above results provide answers to questions posed by Henriksen. It is shown that the class of simple elementary divisor domains coincides with the class of 2-simple Bezout domains. A "possible" reduction of matrices over 3-simple Bezout domain is shown. It is shown that any a right Bezout domain of stable rank 1 is a right 2-Euclidean and any principal ideal domain is Euclidean. In the class of elementary divisor rings a subclass of rings (rings with elementary reduction of matrices) over which any matrix can be diagonalizable only by elementary transformations. It is shown that any commutative 2-Euclidean ring is a ring with elementary reduction of matrices. It is shown that, over any elementary divisor ring, a reduction of matrices up to order 2 can be realized by elementary transformations. Also, a one-sided elementary reduction of non-singular matrices over an adequate ring as well as matrices of special forms over an arbitrary elementary divisor ring is shown.