# How is the addition and multiplication defined in the polynomial ring of infinit. many variables $R[x_i,i\in \mathbb{N}]$

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?

• 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. – Nick Nov 8 at 15:10
• What is the difference the infinitely is not noetherian but the finite one is why? – New2Math Nov 8 at 15:17
• 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. – Nick Nov 8 at 15:19
• Yes that was my next question thank you the first question is now answered – New2Math Nov 8 at 15:22
• If this is your next question, please ask a new question and don't modify this one. – J.-E. Pin Nov 8 at 15:32