Pihura O. Bezout morphic rings

The thesis is devoted to investigation of the morphic rings and finite homomorphic images of commutative Bezout domains, as well as calculations of stable range for the different Bezout rings and their generalizations. Fundamental connections of problems of diagonalization of matrix with stable range and its modern generalizations are obtained. It is proved that any finite homomorphic image of a commutative Bezout domain is a morphic ring, thus answering the question of Nicholson and Sanchez Campos on the existence of morphic rings that are not clean. Necessary and sufficient conditions for the finite homomorphic images of commutative Bezout rings to be Kasch rings are presented; this answers an open question of Faith and Facchini. The stable range of uniquely morphic rings is calculated and it is proved that these rings are elementary divisors rings. We know that in the case of a left quasi morphic ring the property of being uniquely generated is equivalent to that a ring has stable range one. It is proved that for a commutative morphic ring the condition of a neat range one is equivalent to the uniquely generated weak condition up to a neat element. Equivalent definition for the stable range 2 of morphic rings in terms of its Kanfell dimension are found. It is proved that a commutative semiprime Bezout ring is a ring in which zero is an adequate element if and only if it is a regular (von Neuman) ring. Moreover, we answer to the open question of Larsen, Lewis and Shores on the equivalence of rings in which zero is an adequate element and semiregular rings. We define the notion of maximal nongelfand ideal of the commutative Bezout domain, the Gelfand analog of Jacobson radical with their basic properties proved. Answering Zabavsky's question it is proved that any commutative Gelfand local Bezout domain is an elementary divisor ring. It is proved that a commutative domain in which each nonzero prime ideal is a contained in a unique maximal ideal, is a Gelfand local ring. As a consequence we obtain that a commutative Bezout domain in which each nonzero prime ideal is contained in a unique maximal ideal, is an elementary divisor ring. Also the Gelfand local domains and the commutative Bezout domains with finite number of maximal non-Gelfand ideals are studied in the research. It is proved that such rings are the rings of Gelfand range 1 and they are elementary divisor rings.