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I know that if $R$ is a ring $R[x]$ can be defined and it is also a ring with the polynomial multiplication and the addition. In this way the polynomial ring $R[x_1,...,x_n]$ over several variables can be constructed recursively with $R[x_1,...,x_n]:=R[x_1,...,x_{n-1}][x_n]$. How is the polynomial ring of infinitely many variables in the title defined?

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  • $\begingroup$ It is defined exactly the same. Any given polynomial is only allowed to be a FINITE sum. So any calculation you do will only involve finitely many variables and finitely many terms. $\endgroup$ – Nick Nov 8 at 15:10
  • $\begingroup$ What is the difference the infinitely is not noetherian but the finite one is why? $\endgroup$ – New2Math Nov 8 at 15:17
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    $\begingroup$ Your question in the post was only about how addition and multiplication in this ring works. You didn't ask about properties like being Noetherian. $\endgroup$ – Nick Nov 8 at 15:19
  • $\begingroup$ Yes that was my next question thank you the first question is now answered $\endgroup$ – New2Math Nov 8 at 15:22
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    $\begingroup$ If this is your next question, please ask a new question and don't modify this one. $\endgroup$ – J.-E. Pin Nov 8 at 15:32

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