# Question on maximal ideals in an integral domain

1. Aug 6, 2011

### 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?

2. Aug 6, 2011

### Hurkyl

Staff Emeritus
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.

3. Aug 6, 2011

### daveyp225

Ah, yes. I misstepped then. Thanks!