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

Authors

  • B.V. Zabavsky Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
  • O.M. Romaniv Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
https://doi.org/10.15330/cmp.10.2.402-407

Keywords:

Bezout domain, elementary divisor ring, adequate ring, ring of stable range, valuation ring, prime ideal, maximal ideal, comaximal ideal
Published online: 2018-12-31

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 RεiRεi+1 for every 1ir1). 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 aR  the ideal aR a decomposed into a product aR=Q1Qn, where  Qi (i=1,,n) are pairwise comaximal ideals and radQispecR,  is an elementary divisor ring.

How to Cite
(1)
Zabavsky, B.; Romaniv, O. Commutative Bezout Domains in Which Any Nonzero Prime Ideal Is Contained in a Finite Set of Maximal Ideals. Carpathian Math. Publ. 2018, 10, 402-407.