Let $R$ be a ring with unity and assume that $R$ has no nonzero zero-divisors. Let $a,b\in R$, and assume that $ab=1$. Show that $ba=1$, and therefore $a,b$ are units.
I think this question boils down to showing that $R$ is communitive under multiplication (of the ring R), but I don't know how to show it given the conditions. Can someone help please? Thanks
On
First of all, note that $a\neq 0$ (otherwise $ab=0$).
$$ab = 1\ \Rightarrow\ ab -1 = 0 \ \Rightarrow\ aba-a=0\ \Rightarrow\ a(ba-1) = 0 \ \Rightarrow\ ba -1 = 0 \ \Rightarrow\ ba = 1$$
Since $a(ba-1) = 0$ and $a\neq 0$, the other term must be $0$ (i.e. $ba-1=0$). Otherwise, $a$ and $ba-1$ are nonzero zero-divisors.
With your assumptions, $R$ need not be commutative. We can prove the result as follows.
Since $ab=1$ we have that $bab=b$. If it were the case that $ba\neq 1$, then $bab-b=(ba-1)b=0$, and since $ba-1\neq 0$ this contradicts the hypothesis that $R$ has no nonzero zero divisors.