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 $Rε_i\subseteq Rε_{i+1}$ for every $1\le i\le r-1$). We proved that a commutative Bezout domain $R$ in which any nonze\-ro prime ideal is contained in a finite set of maximal ideals and for any nonzero element $a\in R$ the ideal $aR$ a decomposed into a product $aR = Q_1… Q_n$, where $Q_i$ ($i=1,…, n$) are pairwise comaximal ideals and $rad\,Q_i\inspec\, R$, is an elementary divisor ring.