Department of algebra and logic
and participants of seminar
«The problems of elementary divisor rings»
on the successful defense of a PhD thesis
Kuznitska B. M. Restricted rings with adequacy. – On the rights of manuscript.
The thesis for obtaining the Candidate of Physical and Mathematical Sciences degree on the specialty 01.01.06 – algebra and number theory. – Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, 2015.
This thesis introduces the concept of avoidable ring. It is shown that an avoidable ring is a generalization of an adequate ring. And in the case of a commutative Bezout domain is shown that avoidable domains are commutative Bezout domains, finite homomorphic images which are a clean rings. Also it is proved that an avoidable Bezout domain is an elementary divisor domain. It is shown that a commutative Bezout domain of an avoidable range 1 is an elementary divisor ring. The conditions when finite homomorphic images of commutative Bezout domain are semipotent rings are proved. In the class of such domains allocated a class of commutative effective Bezout domains. In particular, it is established their connecting with adequate and avoidable rings. It is shown that a commutative effective Bezout domain is elementary divisor rings. We have calculated stable range of full matrices over a commutative elementary divisor ring. As a consequence, it is shown that full matrices over a commutative elementary divisor ring possess a 2-stage terminating division chain. It is established that regular matrices over a commutative Bezout domain are unit regular matrices.
Keywords: Bezout rings, clean rings, exchange rings, a stable range of elementary divisor rings, adequate rings, avoidable rings, effective rings, semipotent rings.