Combining Euclidean and adequate rings

Authors : Huanyin Chen, Marjan Sheibani

Pages : 506-516

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Publication Date : 9999-12-31

Article Type : Makaleler

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