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Let $k[[x,y]]$ be the ring of formal power series in two variables over a field $k$. A unit in $k[[x,y]]$ is of the form $a_0+f$ where $f\in k[[x,y]]$ and $a_0$ is a unit in $k$. I heard that the quotient of two units in $k[[x,y]]$ is again a unit. How do we show this?

user349424
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    It should be $f\in (x,y)$. If $a,b$ are units then $b=b_0+f$ with $f\in (x,y)$. Then $a/b=ab_0^{-1}(1+b_0^{-1}f)^{-1}=ab_{0}^{-1}\sum_{n=0}^{\infty}(-1)^nb_0^{-1}f)^n$. Since $a$ is also a unit, then $a=a_0+g$ with $g\in (x,y)$. Then $a/b=(a_0+g)b_0^{-1}\sum_{n=0}^{\infty}(-1)^n(b_0^{-1}f)^n=a_0b_0^{-1}+gb_0^{-1}\sum_{n=0}^{\infty}(-1)^n(b_0^{-1}f)^n+a_0b_0^{-1}\sum_{n=0}^{\infty}(-1)^n(b_0^{-1}f)^n+gb_0^{-1}\sum_{n=0}^{\infty}(-1)^n(b_0^{-1}f)^n$. This has the form $a_0b_0^{-1}+h$ with $h\in (x,y)$. – logarithm May 24 '19 at 18:45
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    A quotient of two units in any commutative ring is again a unit.... – Angina Seng May 24 '19 at 19:49

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