Remark 1

The divisibility relation in $\mathbf{K}^*$ depends in an essential way on the particular ring A. If $A = K$ we obtain the « trivial » relation, under which $x \mid y$ for every pair $(x, y)$ of elements of $\mathbf{K}^*$. Let p and q be prime numbers; the rational numbers r/s whose denominators are not multiples of $p$ (resp. q) form a subring $\mathbf{Z}_{(p)}$ (resp. $\mathbf{Z}_{(q)}$) of $\mathbf{Q}$; the divisibility relations in $\mathbf{Q}^*$ relative to these two rings are distinct if $p \neq q$, the number $p/q$ being a multiple of 1 under one relation but not under the other.