Some really bad rings
by math_explorer, Sep 26, 2017, 12:36 AM
Consider ![$\mathbb{Q}[x_1, x_2, x_3, \ldots]$](//latex.artofproblemsolving.com/a/4/e/a4e4266f8010c8ef30cab8ea7aba4bd45d11b5d8.png)
where you adjoin infinitely many free variables. This has infinite Krull dimension because ideals 
are all prime. It's also local; the ideal generated by all the variables is the unique maximal ideal.
Consider![$\mathbb{Q}[x, x^{\omega-n}\text{ for all }n \in \mathbb{Z}]$](//latex.artofproblemsolving.com/e/4/9/e49d2dc455fa0b66df6efb95c72a6007cce6e850.png)
; think of 
like an infinity. It is local because 
is the unique maximal ideal. The ideal generated by all 
is prime; that maximal ideal divides it infinitely many times.
Consider![$\mathbb{Z} + x\mathbb{Q}[x]$](//latex.artofproblemsolving.com/9/8/e/98ebc9f23f78305d98bd7816632348f36b065aba.png)
, the ring of polynomials with integer constant coefficient. Each integer prime or prime integer or something 
generates a maximal ideal 
. The intersection of those infinitely many maximal ideals is the prime ideal .
![$\mathbb{Q}[x_1, x_2, x_3, \ldots]$](http://latex.artofproblemsolving.com/a/4/e/a4e4266f8010c8ef30cab8ea7aba4bd45d11b5d8.png)
where you adjoin infinitely many free variables. This has infinite Krull dimension because ideals 
are all prime. It's also local; the ideal generated by all the variables is the unique maximal ideal.Consider
![$\mathbb{Q}[x, x^{\omega-n}\text{ for all }n \in \mathbb{Z}]$](http://latex.artofproblemsolving.com/e/4/9/e49d2dc455fa0b66df6efb95c72a6007cce6e850.png)
; think of 
like an infinity. It is local because 
is the unique maximal ideal. The ideal generated by all 
is prime; that maximal ideal divides it infinitely many times.Consider
![$\mathbb{Z} + x\mathbb{Q}[x]$](http://latex.artofproblemsolving.com/9/8/e/98ebc9f23f78305d98bd7816632348f36b065aba.png)
, the ring of polynomials with integer constant coefficient. Each integer prime or prime integer or something 
generates a maximal ideal 
. The intersection of those infinitely many maximal ideals is the prime ideal .
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