Answer by i. m. soloveichik for A question about Euclidean Domain
Try an element $c$ of smallest value greater than $1$. Then division by $c$ has a remainder of zero or value $1$, which is therefore an unit.
View ArticleAnswer by Arturo Magidin for A question about Euclidean Domain
If $R$ is not a field, then it has nonunits. Consider the set $S=\{\varphi(a)\mid a\text{ is not a unit, and }a\neq 0\}$, where $\varphi$ is the Euclidean function.It is a nonempty set of positive...
View ArticleA question about Euclidean Domain
This is a problem from Aluffi's book, chapter V 2.17."Let $R$ be a Euclidean Domain that is not a field. Prove that there exists a nonzero, nonunit element $c$ in $R$ such that $\forall a \in R$,...
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