A ring is a field if and only if its only prime ideal is (0)
3129
A commutative ring is a field iff the only ideals are $(0)$ and $(1)$
69710
Is it true that an ideal is primary iff its radical is prime?
134773
A domain is a field if it has a common multiple $\!\neq\! 0$ of all elements $\!\neq\! 0$
94174
Ring of polynomials over a field has infinitely many primes
6646
A (non-artificial) example of a ring without maximal ideals
45962
Show that any prime ideal from such a ring is maximal.
70776
Characteristic of a field is $0$ or prime
53770
Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime.
103830
Each element of a ring is either a unit or a nilpotent element iff the ring has a unique prime ideal
107463
Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime?
21604
A complete ring with respect to an ideal
115920
Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
72230
Equivalence of definitions of prime ideal in commutative ring
135781
Division ring if and only if it has no proper left ideals.
69611
If $R$ and $S$ are rings then $R \times S$ is never a field
67669
Prime ideals of a finite direct product ring
138130
If $I$ is a maximal ideal of $R$, why is $R/I$ a field?