# Homework: If a is a unit, prove that b divides c if and only if ab divides c.

As it says in the title, this is a homework question, so try not to give everything away. I'm just looking for a starting point. The question is stated as follows:

Let $R$ be a commutative, unital ring. Recall that $a\in R$ is a unit, if $a$ has a multiplicative inverse in $R$. Also, recall that for $b,c\in R$ we say $b$ divides $c$, if there exists some $d\in R$ such that $c=bd$.

Let $a$ be a unit. Prove that $b$ divides $c$, if and only if $ab$ divides $c$.

## 1 Answer

$b\,\vert\, c\Leftrightarrow \exists x\in R$ such that $c=bx$. Since $a$ is a unit, then $c=abxa^{-1}$. Let $y=xa^{-1}\in R$ and observe that $c=aby$. Thus $ab\,\vert\, c$.

A similar argument should give you the reverse direction.

You will need to supply some details from my forward direction, but it should be straight forward.

• Thank you. I was actually considering this as my solution, but thought it was wrong for some reason. – user131736 Apr 2 '14 at 4:39