Does the ring of polynomial of infinitely many variables $\mathbb{R} [ x_1 , x_2 , \ldots ]$ have one maximal ideal or infinitely many?

Question

I know that $R:= \mathbb{R} [x_1 , x_2 , \ldots ]$ is non Noetherian, i.e. there is a increasing chain of ideals $\{ I_n \}_{ n \in \mathbb{N} }$such that $I_n \subsetneq I_{n+1}$ for all $n \in \mathbb{N}$ and furthermore, I know that given any ideal $I$, we can find a maximal ideal $M$ such that $I \subset M$.

Can I use this to answer this? Can someone drop me some hints?

Answer 1

Hint: $\mathbb R[x_1]$ is a quotient, and the maximal ideals in this quotient correspond to some of those in your big ring. Does $\mathbb R[x_1]$ have one maximal ideal or does it have infinitely many?

Answer 2

There are infinitely many maximal ideals. Indeed, for any real number $r$, the ideal $(x_1+r,x_2,x_3,x_4,...)$ is maximal (of course, $r$ could be added to other variables also, while still yielding a maximal ideal).