[Math] Why are irreducible elements non-units

Question

I know this may seem trivial but I'm trying to grasp why irreducible elements are non-units. If an element p is a unit and b is its inverse, then $pb = 1, \forall p,b \in R$, R is a ring. Does this imply that b is a factor of p, thus making it reducible?

Answer 1

Here is another example: suppose your teacher instructs you to factor:

$p(x) = x + 1$ in the rational number system.

You might reply, "it's already factored", but then your teacher says:

"No, $p(x) = 2\cdot \left(\dfrac{x}{2} + \dfrac{1}{2}\right)$".

You realize, with a sinking feeling, you'll never "be finished factoring". Intuitively, factoring ought to stop "at some basic level". Factoring "up to units" IS that basic level, an opt-out that allows us to finally be "done". (note that $2$ is a unit in $\Bbb Q$, so the two factorizations above are "really the same").