Characterize units in formal power series $R[[x]]$

Question

Suppose $R$ is a commutative ring with unity. Define $R[[x]]$ as "formal power series in the variable $x$ with coefficients from $R$". These are the infinite sums of the form $ \sum_{n=0}^\infty a_ix^i, a_i\in R. $ Is there any way to characterize all of the units in this ring $R[[x]]$?

Answer 1

The units of $R[[x]]$ are exactly the formal power series whose constant term correspond to units of $R$. One way to prove this is to consider truncations of the power series and show that as $\deg(p(x))$ increases, you can eventually find a $q(x))$ of the same degree such that all the terms of $p(x)q(x)$ less than $K$ and note that the constant term are $0$. Thus the nonzero terms get "pushed off" the power series, and leave us with the product of the constant terms. Another way to see this is to note that you can find a $q$ such that $p(x)q(x)-p_0q_0$ has a root at every element of the ring. Once you've done this, it's simple to see that this product can be $1$ iff $p_0$ is a unit.