$R$ is a subring of a field $F$ with $x\in R$ or $x^{-1}$ for all $x\in F^\times$. If $I,J$ are ideals of $R$, then $I\subseteq J$ or $J\subseteq I$.
3936
Characterization of valuation rings
29019
Ideals in direct product of rings
45684
Is quotient of a ring by a power of a maximal ideal local?
135682
Are distinct prime ideals in a ring always coprime? If not, then when are they?
138752
Prove the Radical of an Ideal is 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
65091
Subring which is not an ideal?
36996
If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.
76813
The ring $ℤ/nℤ$ is a field if and only if $n$ is prime
80961
Prove that a commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain
50445
Explaining the product of two ideals
58425
Defining the Product of Ideals
22544
Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain.
72230
Equivalence of definitions of prime ideal in commutative ring
141100
Nontrivial subring with unity different from the whole ring?
30714
A question on ring homomorphisms and maximal ideals.
87103
Quick question about ideals of field $F$ in $F[x]$ are principal.