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

Irwing Kaplansky

Adequate properties of the elements with almost stable range 1 of a commutative elementary divisor domain

Let R — commutative elementary divisor domain which is not a ring of stable range 1. Then exist element a∈R with almost stable range 1 (i.e. for any elements b,c∈R such that aR+bR+cR=R exist elements r that aR+(br+c)R=R). In this report we describe algebraic properties these elements r∈R. Lets note that a commutative elementary divisor domain of stable range 1 is 2-Euclidean domain of stable range 1.

Almost zip Bezout domain

Theorem. Let R be a commutative Bezout domain and in R for any non-zero and non-unit element a factor-ring R/rad(aR) is zip. Then R is 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
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Scientific seminar, August, 2017