Commutative Bezout domains in which any nonzero prime ideal is contained in a finite set of maximal ideals
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 2
Abstract
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(ε1,ε2,…,εr,0,…,0) such that PAQ=D and Rεi⊆Rεi+1 for every 1≤i≤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∈R the ideal aR a decomposed into a product aR=Q1…Qn, where Qi (i=1,…,n) are pairwise comaximal ideals and radQi∈specR, is an elementary divisor ring.
Authors and Affiliations
B. V. Zabavsky, O. M. Romaniv
Keywords
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- EP ID EP535565
- DOI 10.15330/cmp.10.2.402-407
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