Krull's theorem and uniqueness Commutative rings with unit must have a maximal ideal by Krull's theorem.
But is it true, in general, that such sets must have a unique maximal ideal?
Does it matter if the ring is finite or infinite?
 A: The "result due to Krull" about rings with identity always having maximal ideals is true (with the same proof) even for noncommutative rings, and moreover, the same sort of proof shows rings have maximal right ideals and maximal left ideals.
Andrew mentioned in the comments that commutative rings with identity having a unique maximal ideal are called local rings, and these are indeed a very important class of rings. However, certainly not all rings are local. $\mathbb{Z}$ has multiple maximal ideals, one corresponding to each prime element of $\mathbb{Z}$.
Actually in noncommutative ring theory, we also talk about local rings. The defintion is a little different though: "has a unique maximal right ideal." This turns out to be equivalent to "has a unique maximal left ideal," and it implies that the ring has a unique maximal ideal. However there are noncommutative rings with unique maximal two-sided ideals which do not have unique maximal right ideals. An example would be an $2\times 2$ matrix ring over a field, which has maximal ideal $\{0\}$, but which has more than two maximal right ideals.
