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Let $A = {\Bbb F}_p[[X_1,X_2,...]]$

be the ring of formal power series with infinitely many variables over the finite field ${\Bbb F}_p.$

$A$ consists of such formal sum elements as $\sum c_{e_1,...,e_n}X_1^{e_1}・・・X_n^{e_n}$

with $c_{e_1,...,e_n} ∈ {\Bbb F}_p.$

For example, $X_1 + X_2 +\dots \in A.$

Question: Is $A$ coherent ?

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    $\begingroup$ The notation is misleading: is $X_\infty$ a variable? Also, you should clarify what you mean by "power series ring with infinitely many variables" since there are several possible definitions (e.g. sometimes the series $X_1+X_2+X_3+\dots$ is allowed and sometimes it is not). $\endgroup$
    – Matthieu Romagny
    Apr 25, 2016 at 6:43
  • $\begingroup$ As it was written $A= F_p[[X_1,X_2,...]]$ and "the ring...", there is nothing to suppose, so I guessed a subset is meant. If not, please modify. $\endgroup$
    – Wolfgang
    Apr 25, 2016 at 14:40
  • $\begingroup$ @Wolfgang: I think the issue raised by Matthieu Romagny is whether we are talking about the ring consisting of formal series with only finitely many nonzero terms, or the ring of all formal series. $\endgroup$
    – Lazzaro Campeotti
    Apr 25, 2016 at 14:48
  • $\begingroup$ What I meant is that there is no canonical definition of a ring of formal power series in countably many variables. There are several candidates, each with its own interest depending on the situation. One natural candidate is the completion of the ring of polynomials $F_p[X_1,X_2,\dots]$ with respect to the maximal ideal $(X_1,X_2,\dots)$. Another useful candidate is that introduced in Bourbaki, Algèbre, chapitre IV, whose elements are arbitrary formal sums of monomials $a_iX^i$ with $i$ a finite tuple of... $\endgroup$
    – Matthieu Romagny
    Apr 26, 2016 at 13:16
  • $\begingroup$ ... natural integers. In the former ring, the element $X_1+X_2+\dots$ is not allowed; in the latter it is. $\endgroup$
    – Matthieu Romagny
    Apr 26, 2016 at 13:16

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