Bokhonka V. Stable range and its generalization in Bezout rings.

Introduce the concept of neat range one for duo-Bezout domains. Show that distributive Bezout domain is an elementary divisor ring if and only if it is a duo-domain of neat range one. Shown that the Bezout stable range one domain with Dubrovin's condition and the condition Z is an elementary divisor ring. Introduces the de nition of gelfand and nongelfand elements and ideals, we establish sets of properties elements or ideals, consider gelfand analogue of Jacobson radical. Proved that any locally gelfand Bezout duodomain is an elementary divisor ring. Introduces the concept of ring Toeplitz and ring unit square stable range one. Establishes conditions when nite homomorphic images commutative Bezout doamins is a semipotent rings.7452