# Some properties of almost pm-elements and elementary divisor rings

- Written by A. Gatalevych
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*Associate Professor**Department of Algebra and Logic** Faculty of Mechanics and Mathematics** Ivan Franko National University of L'viv*

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Abstract:

*A ring R is called pm-ring if every prime ideal of R is contained in a unique maximal ideal. An element a of a ring R is called almost pm-element if factor ring R/aR is a pm-ring. *

*By S _{0} we denote the set of all almost pm-elements and by and U(R) the set of units of a ring R. *

**Theorem 1.*** S _{0} is a saturated and multiplicatively closed set. *

**Theorem 2.** Let R be a commutative Bezout domain and U(R)=S_{0}. Then R is an elementary divisor ring iff stable range of R is equal one.

**Theorem 3.** Let R be a commutative Bezout domain and for every a∈ R\(0) the stable range of a ring R/aR is greater then one. Then R is not an elementary divisor ring.

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