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Let $R$ be a commutative ring with 1. Is there a nice way to characterise elements $$f(T) = a_0 + a_1 T + a_2 T^2 + \dots \in R[[T]]$$ whose inverses are polynomials? Something like "$f$ has invertible constant term $and$..."?

If it helps, the ring $R$ can be specialised to any field $K$. A crude guess was that maybe $f$'s coefficients have to be periodic in some way, but examples like $f(T) = 1/(1+T)^2$ show that's false.

It seems natural to me that such a nice property should have a correspondingly nice bearing on the coefficients of $f$, but maybe this is too much to hope for?

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    Please note that I've asked for the inverse of $f(T)$ to be a polynomial, not just a power series. So your condition is merely necessary. – 2017-01-25
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    @Mr.Chip You might be interested in [my colleague's 2009 dissertation](https://etd.ohiolink.edu/rws_etd/document/get/ohiou1257886601/inline) on rational and periodic powerseries. – 2017-01-25
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    That looks very useful; thanks rschwieb. – 2017-01-25
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    I suggest you to emphasize that the ring is unital (and commutative?). In the case that $f$ is already a polynomial, (is is well-known that) it is sufficient and necessary (in the commutative case; I am very ignorant about the noncommutative case... shame on me!) the constant term to be invertible and the others to be nilpotent. Nice question, I will give a thought. – 2017-01-25
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    Very sorry about missing the condition of the inverse needing to be a polynomial. Deleting the previous comment. If $R$ were a finite field, then every polynomial $p(T)$ with a non-zero constant term is a factor of some binomial $T^m-1$, and this means that $1/p(T)$ would be periodic. This process can also be inverted in the sense that a periodic power series is a rational function. Don't know much else. – 2017-01-25
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    The more natural question is to ask which power series are rational functions, and there the answer, at least over a field, is that they are precisely the power series whose coefficients satisfy linear recurrences (and various more specific characterizations are possible). In this special case you get linear recurrences with some specific initial conditions. – 2017-01-25

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