Proof of a Property of a Monoid: Show that the following are equivalent

Question

I am stuck on the following question:

Show that the following are equivalent for a monoid M:

1. If $ab$ is a unit, then both $a$ and $b$ are units
2. If $ab = 1$, then $ba = 1$

I am able to show that (2) $\Rightarrow$ (1) as follows:

Suppose $ab$ is a unit. Denote its inverse as $ k = (ab)^{-1}$.

Now, consider: $(ab)k=1 \Leftrightarrow a(bk)=1 \Rightarrow (bk)a=1$ , by (2)

So, we have that: $a(bk) = 1 = (bk)a$, hence $a$ is a unit. In a similar fashion, $b$ is also a unit. Thus, we have proven that (2)$\Rightarrow$(1)

But I am having trouble with (1)$\Rightarrow$(2), namely, if I assume $ab=1$, how can I use (1) to continue the proof?

Thank you in advance for you help!

Answer 1

Notice that $1$ is itself a unit. Now assume that $ab=1$; this immediately satisfies the hypothesis of $(1)$, so we can conclude that both $a$ and $b$ are units. Then \begin{align*} &ab=1\\ \implies &abb^{-1}=b^{-1}\\ \implies &a=b^{-1}\\ \implies &ba=bb^{-1}\\ \implies &ba=1 \end{align*}