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

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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

DOI:

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

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 $D=\mathrm{diag}(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_r,0,\dots,0)$ such that  $PAQ=D$  and $R\varepsilon_i\subseteq R\varepsilon_{i+1}$ 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 $aR = Q_1\ldots Q_n$, where  $Q_i$ ($i=1,\ldots, n$) are pairwise comaximal ideals and $\mathrm{rad}\,Q_i\in\mathrm{spec}\, R$,  is an elementary divisor ring.

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Published

2018-12-31

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: Array. Carpathian Math. Publ. 2018, 10, 402-407.

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Scientific articles