Elementary Divisor Rings and Finitely Presented Modules

Conditions for stable range of an elementary divisor rings

“…Since by Theorem 1 and Proposition 1 a commutative Bezout P M *domain is a domain of Gelfand range 1, as a consequence of Theorem 3 we have the following result which is answer to open questions raised in [5,12]. An obvious example [13] of a local Gelfand ring is the Henriksen example [3] R = {z 0 + a 1 x + a 2 x 2 + ... + a n x n + ...|z 0 ∈ Z, a i ∈ Q}.…”

“…Definition of an elementary divisor ring was given by I. Kaplansky in 1949 [4]. Since any elementary divisor ring is a Bezout ring, we obtain the question: whether an arbitrary Bezout ring is an elementary divisor ring [3,4,5]. Gilman and Henriksen [2] constructed an example of a commutative Bezout ring which is not an elementary divisor ring.…”

Elementary matrix reduction over certain rings

“…As it is a finite ring, for every element s of Z n , we see that Z(s), i.e., the set of maximal ideals containing a, is finite. In light of [4,Theorem 4.3], Z n is adequate, and we are through.…”

On the Weak Grothendieck Group of a Morphic Ring and its Representations

“…0 (R) and the usual Grothendieck group K 0 (R) coincide. The main motivation of these investigations is that in [11]it is proved that a commutative Bezout domain is an elementary divisor ring if and only if any quotient ring R/aR is so, where a is an arbitrary nonzero element of R. Since any finite homomorphic image of a commutative Bezout domain R is a morphic ring [14] then the studies of the ring K ′ 0 (R/aR) become related to the famous elementary divisor ring problem [8].…”