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I am studying ring of formal power series over $\mathbb{R}$ and trying to solve this following exercise.

The ring of formal power series over $\mathbb{R}$, denoted $\mathbb{R} [[ x ]]$, is a ring whose elements are power series $a_0 + a_1 x + a_2 x^2 + \ldots$ with each $a_i \in \mathbb{R}$, without regard to whether or not such series converge or eventually terminate. The addition and multiplication of series is defined in the natural way.

Is there any not invertible element in $\mathbb{R} [[ x ]]$?

Is any unit of $\mathbb{R} [[ x ]]$ is of form $\sum_{i=0}^{\infty}a_ix^i$ where $a_i$ are invertible in $\mathbb{R}$?

If this is true, then all the element in $\mathbb{R} [[ x ]]$ is true since $\mathbb{R}$ is a field.

Is $\mathbb{R} [[ x ]]$ is a local ring?

To prove $\mathbb{R} [[ x ]]$ is a local ring, I need to prove that there is a unique maximal ideal. However I am unable to find one as of now.

Any help would be greatly appreciated.

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    $\begingroup$ Is $x$ invertible? Which $a_i$ are not invertible in $\mathbb{R}$? What characterizations of local rings do you know? $\endgroup$
    – user113102
    Nov 6, 2019 at 15:59

1 Answer 1

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Hint 1 : take an invertible element $A=\sum_{i\geqslant 0} a_i X^i$, and let $B=\sum_{i\geqslant 0} b_i X^i$ be the multiplicative inverse of $A$. One has $1 = AB = \sum_{i\geqslant 0} \big( \sum_{k=0}^i a_k b_{i-k} \big) X^i$. Compare termwise, eg you get $a_0b_0=1$, $a_0b_1 + a_1 b_0=0$ ...

Hint 2 : a ring is local iff the set of all non-invertible elements forms an ideal.

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