chain of prime ideals in polynomial ring $S=\Bbb{R}[x_1,x_2,...,x_n,...]$

Question

Consider the polynomial ring of countable variables with coefficients in the real numbers, i.e, $S=\Bbb{R}[x_1,x_2,x_3,...,x_n,...]$. Is the following question true?

Question: Is there any maximal ideal in $S$ such that it contains a chain with uncountable number of prime ideals?

PS: We recall that a set $\{P_{\alpha}\} _{\alpha \in T}$ is a chain of prime ideals if for each $\alpha , \beta \in T$, either $P_{\alpha} \subseteq P_{\beta}$ or $P_{\beta} \subseteq P_{\alpha}$.

PPS: By this notation, i need a chain $\{P_{\alpha}\} _{\alpha \in T}$ of prime ideals in $S$ for which $T$ is uncountable.

Answer 1

Pick your favorite bijection $\phi$ from the natural numbers to the rational numbers. For each real number $\alpha$, let $I_\alpha$ be the ideal generated by all $x_n$ with $\phi(n)<\alpha$.

Then $I_\alpha\subset I_\beta$ if and only if $\alpha\le \beta$, so the $I_\alpha$ form an uncountable chain, clearly contained in the maximal ideal generated by all of the $x_i$.