In ring theory a given ring $R$ is called a Euclidean domain if there exists a function $\sigma:R -\{0\}\rightarrow \{0,1,2,3...\} $ which satisfies the division algorithm i.e.
$ $ if $a,b \in R$ then there exists $q,r \in R$ such that $b=aq+r$ and either $r=0$ or $\sigma(r)\lt \sigma(a) $
Now I want to ask if we can prove, using just this definition that an element of larger degree won't divide an element of smaller degree. In specific rings such as
(i) the integers
we can say$$\sigma(a)=|a|$$ $$\sigma(ab)=\sigma(a)\sigma(b)$$
ii) polynomials
where$$\sigma(f(x))=deg(f(x))$$$$\sigma(ab)=\sigma(a)+\sigma(b)$$ So in both the above cases the size of product of two elements will always be greater than or equal to the size of individual elements and hence the larger element can never divide the smaller element. But is this true in general for all Euclidean domains ? And how will we prove that