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