If $A$ and $B$ are proper ideals of a ring with no zero divisors, show that $A\cap B \neq \{0\}$
74559
$A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.
41629
Structure of ideals in the product of two rings
116883
About ideals in a ring which is a $k$ vector space
69353
Show that a ring with only trivial right ideals is either a division ring or $|R|=p$ and $R^2=\{0\}$. Why would $R$ be finite?
114343
If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$
117786
Show that some monomial ideal is primary
587
Ideal Generated by the Union of Two Ideals
1141
Existence of totally ordered ring with zero divisors
114193
On the absolute norm of an ideal
99715
Proper ideal and units proof
21840
Zero divisors and units for the ring $\mathbb{Z}/n\mathbb{Z}$ with integer $n$
45962
Show that any prime ideal from such a ring is maximal.
146304
prove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero
62917
Associated Prime Ideals in a Noetherian Ring; Exercise 6.4 in Matsumura
138826
To show that either $R$ is a field or $R$ is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$.
34203
if $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring.
138672
Finding Ideals in $\begin{bmatrix} \mathbb{Q} & \mathbb{Q}\\ 0 & 0 \end{bmatrix}$