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.



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


