Question on maximal ideals in an integral domain

[daveyp225]

In an integral domain, I found that the number of maximal ideals in a Notherian ring containing a particular element is finite. If the condition is dropped that the ring be Notherian, can anything like this be said?

 

[Hurkyl]

If k is a field, then the polynomial ring k[x,y] is a Noetherian integral domain. (regular too) However, the element x is contained in infinitely many maximal ideals; for example:

• (x, y)
• (x, y-1)
• (x, y-2)

More generally, a maximal ideal of this ring contains x if and only if it is of the form (x, f(y)) for some non-constant polynomial f that is irreducible over k.

 

[daveyp225]

Ah, yes. I misstepped then. Thanks!