Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals - Zabavsky, B.V.; Romaniv, O.M. (Carpathian Mathematical Publications, 2018)
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 R is called an elementary divisor ring if every matrix over R has a canonical diagonal reduction (we say that a matrix A over R has a canonical diagonal reduction if for the matrix A there exist invertible matrices P and Q of appropriate sizes and a diagonal matrix D=diag(ε_1,ε_2,…,ε_r,0,…,0) such that PAQ=D and Rε_iRε_i+1 for every 1ir-1). We proved that a commutative Bezout domain R in which any nonzero prime ideal is contained in a finite set of maximal ideals and for any nonzero element aR the ideal aR a decomposed into a product aR = Q_1… Q_n, where Q_i (i=1,…, n) are pairwise comaximal ideals and rad Q_ispec R, is an elementary divisor ring.