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Let $R$ be a commutative ring and define $$B = \{ f = \sum_{n=0}^{\infty} a_n X^n: a_n \in R\}$$ where $X$ is an indeterminate over $R$.

a. Show that $B$ is a ring.

b. Prove or disprove: $f$ is a unit if and only if $a_0$ is a unit.

I could not relate this question with the post Characterizing units in polynomial rings exactly since there is nothing about nilpotent elements. Maybe I need a more detailed explanation.

silent_man
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1 Answers1

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Hint:

$\rightarrow$ is easy. To prove $\leftarrow$, you may suppose $a_0=1$.

Just write the (infinite) system of equations for the coefficients of $c_0+c_1 X+\dots+c_nX^n+\dotsm $ to be an inverse of $f$: \begin{align} c_0&=1, &c_0a_1+c_1&=0,& &c_0a_2+c_1a_1+c_2=0,\\ \dots&\dots\dots\dots&&&& c_0a_n+c_1a_{n-1}+\dots+c_{n-1}a_1+c_n=0,\\ \dots&\dots\dots\dots \end{align} We thus have a recursive definition of the coefficients: $$c_0=1,\quad c_n=-\sum_{k=0}^{n-1}c_ka_{n-k}.$$

Bernard
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