Does any natural number q divide 0 because there exists a natural number m such that qm=0 (m=0)?
Is it true that 0 does not divide any natural number >0, because there does not exist a number q such that 0*q=m where m is any natural number >0.
What about when m is 0. does 0 divide 0, is it true that the definition above holds but it is also undefined? thanks
Welcome to MSE! As a general hint, let $R$ be ring. Then for each ring element $r$, $r0 = 0 = 0r$, since $r0 = r(r+(-r)) = r^2-r^2 = 0$; similarly, $0r=0$. So $0$ is absorbing in every ring.
If you look at the relation of divisiblity on the monoid of natural numbers, $a\mid b$ ($a$ divides $b$) means that $a\cdot c=d$ for some number $c$. Here $0$ is the largest element, since $a\mid 0$ by taking $c=0$ and $1$ is the smallest element, since $1\mid b$ by taking $c=b$.
