Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals

Keywords:
Bezout domain, elementary divisor ring, adequate ring, ring of stable range, valuation ring, prime ideal, maximal ideal, comaximal idealAbstract
We investigate commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals. In particular, we have described the class of such rings, which are elementary divisor rings. A ring is called an elementary divisor ring if every matrix over has a canonical diagonal reduction (we say that a matrix over has a canonical diagonal reduction if for the matrix there exist invertible matrices and of appropriate sizes and a diagonal matrix such that and for every ). We proved that a commutative Bezout domain in which any nonze\-ro prime ideal is contained in a finite set of maximal ideals and for any nonzero element the ideal a decomposed into a product , where () are pairwise comaximal ideals and , is an elementary divisor ring.