Turkish Journal of Mathematics
DOI
10.3906/mat-1502-58
Abstract
We combine Euclidean and adequate rings, and introduce a new type of ring. A ring $R$ is called an E-adequate ring provided that for any $a,b\in R$ such that $aR+bR=R$ and $c\neq 0$ there exists $y\in R$ such that $(a+by,c)$ is an E-adequate pair. We shall prove that an E-adequate ring is an elementary divisor ring if and only if it is a Hermite ring. Elementary matrix reduction over such rings is also studied. We thereby generalize Domsha, Vasiunyk, and Zabavsky's theorems to a much wider class of rings.
Keywords
Euclidean rings, adequate rings, elementary divisor rings, elementary matrix reduction
First Page
506
Last Page
516
Recommended Citation
CHEN, HUANYIN and SHEIBANI, MARJAN
(2016)
"Combining Euclidean and adequate rings,"
Turkish Journal of Mathematics: Vol. 40:
No.
3, Article 3.
https://doi.org/10.3906/mat-1502-58
Available at:
https://journals.tubitak.gov.tr/math/vol40/iss3/3
