Bohdan Zabavsky, "Diagonal reduction of matrices over rings"

Bohdan Zabavsky, Lviv University, Ukraine

Diagonal reduction of matrices over rings

The monograph is devoted to the presentation of the theory of reduction of matrices to diagonal form over rings of finite stable rank. The stable rank of rings is a working tool of all investigations. Mainly, the monograph examines the rings of finite stable rank, Bezout rings and elementary divisor rings. We introduce several new concepts (n-Hermitian ring, maximal non-principal ideals, ring with elementary reduction matrices), which work successfully.
Basically, all presented results belong to the author and are exposed for the first time.
The book can be useful to both experts in algebra and in the algebraic K-theory.

Zabavsky B.V. Diagonal reduction of matrices over rings // Mathematical Studies, Monograph Series, volume XVI, VNTL Publishers, 2012. Lviv. 251 pp.


Finite stable range rings

n-Hermite rings
Commutative rings with compact space of minimal prime ideals
Diagonalization of matrices over von Neumann regular rings of finite stable range
Direct finiteness of matrix rings
Cases when Hermite quotient ring implies Hermite main ring
Stable range 1 rings

Bezout rings

Maximal nonprincipal ideals
Adequate rings and their generalizations
Maximal and prime spectrum of commutative Bezout rings without zero divisors

Elementary divisor rings

Simple elementary divisor rings
Elementary matrix reduction
Matrix reduction over locally countable commutative Bezout rings
Matrix reduction over Bezout ring with unique maximal nonprincipal ideal that satisfies Dubrovin condition
"Weak" diagonal reduction of matrices over von Neumann regular rings
Elementary divisor rings with L condition
Matrix reduction over almost atomic Bezout rings without zero divisors of stable range 1 that satisfies Dubrovin condition
Matrix reduction over commutative Bezout rings without zero divisors of stable range 1 in localizations
Simultaneous reduction of pair of matrices to special triangular form over everywhere adequate ring

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