Let $R$ be a commutative ring with $1\neq 0$. Prove that if every proper ideal of $R$ is prime, then $R$ is a field.
70751
A commutative ring whose all proper ideals are prime is a field.
108380
Assume $P$ is a prime ideal s.t. $K \subset P$, show $f(P)$ is a prime ideal
133878
Ring of order $p^2$ is commutative.
94157
Normal Ring and Prime Ideal whose Square is Principal
67669
Prime ideals of a finite direct product ring
18971
Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
74559
$A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.
117536
Why is the localization at a prime ideal a local ring?
135586
Prove that the preimage of a prime ideal is also prime.
75558
Counter-example for prime ideals in a non commutative ring.
85206
Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.
29019
Ideals in direct product of rings
130466
Proving every prime ideal is maximal in a finite dimensional ring over a field.
74370
$P$ is a prime ideal of $R$ iff $R/P$ is an integral domain. : $P≠R$
131943
How to directly prove that $M$ is maximal ideal of $A$ iff $A/M$ is a field?
14055
How to show that a finite commutative ring without zero divisors is a field?
72230
Equivalence of definitions of prime ideal in commutative ring