A principal maximal ideal

Question

Let $(R,m)$ be a local integral domain, and $t\in m^{-1}$ be such that $tm=R$. Is it true that $m$ is principal?

If $t=a/b$ with $a,b\in R$ and $b\not =0$ then $ac/b=1$ for some $c\in m$, and since $R$ is a domain, we have $b=ac\in m$. Now, if $x\in m$ is arbitrary then $ax/b=y\in R$, so $x=(y/a)b$. If we could prove that $y/a\in R$ we are done.

Thanks for any answering!

Answer 1

From $tc=1$ we get $t=\frac 1c$. Then $\frac 1c\mathfrak m=R\Rightarrow\mathfrak m=cR$ and you are done.