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 RR is called an elementary divisor ring if every matrix over RR has a canonical diagonal reduction (we say that a matrix AA over RR has a canonical diagonal reduction if for the matrix AA there exist invertible matrices PP and QQ of appropriate sizes and a diagonal matrix D=diag(ε1,ε2,…,εr,0,…,0)D=diag(ε1,ε2,…,εr,0,…,0) such that PAQ=DPAQ=D and Rεi⊆Rεi+1Rεi⊆Rεi+1 for every 1≤i≤r−11≤i≤r−1). We proved that a commutative Bezout domain RR in which any nonze\-ro prime ideal is contained in a finite set of maximal ideals and for any nonzero element a∈Ra∈R the ideal aRaR a decomposed into a product aR=Q1…QnaR=Q1…Qn, where QiQi (i=1,…,ni=1,…,n) are pairwise comaximal ideals and radQi∈specRradQi∈specR, is an elementary divisor ring.