On a Generalization of the Euclidean Algorithm

Abstract

In any valuated ringR, more specifically, a ring withadegree-functiond(a) as defined in §2, the notion of rightR-dependence may be denned as follows: (i) The elementsa1, …,arofRareright R-dependentif there existb1, …,br∈Rsuch that. (ii) An elementaofRis said to beright R-dependenton the elementsa1, …,arofRifa= 0 or if there existc1, …,cr∈Rsuch that.