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"I liked the algebraic way of looking at things.
I’m additionally fascinated when the algebraic method is applied to infinite objects”.

Irwing Kaplansky

Some properties of almost pm-elements and elementary divisor rings

 

 

Andry Gatalevych

Associate Professor
Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv

 
 

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 S0 we denote the set of all almost pm-elements and by and U(R) the set of units of a ring R.

Theorem 1. S0 is a saturated and multiplicatively closed set.

Theorem 2. Let R be a commutative Bezout domain and U(R)=S0. 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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Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
1 Universytetska Str., 79000 Lviv, Ukraine
Tel: (+380 322) 394 172
E-mail: oromaniv at franko.lviv.ua

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