A ring $R$ is a field $\iff R$ has exactly two ideals.
3129
A commutative ring is a field iff the only ideals are $(0)$ and $(1)$
37393
$R$ is commutative, $I$,$J$ are ideals, $I+J=R$, then $IJ=I\cap J$
124564
Prove the ring $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ has inverse and is a field
90314
Maximal ideals in $R[x]$
20877
The set of all nilpotent elements is an ideal
24247
Ring Inside an Algebraic Field Extension
81296
If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
115920
Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
117590
Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal
47134
Ideals generated by two elements
62757
Showing that a ring is a field as well for one of the provided choices.
121750
Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.
79957
If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors
11573
Rings and some of their properties
136654
$(x_1-a_1, x_2-a_2)$ is a maximal ideal of $K[x_1,x_2]$
148208
In Noetherian ring $R$, ideal $I\subset R$, $P\in\mathrm{Ass}(I),P=I:a$ for some $a\in R$.
46047
Check whether the quotient ring $R:=\mathbb{C}[x,y,z]/(xy-z^2)$ is a field or an integral domain