if $S$ is a ring with no proper left ideals, then either $S^2=0$ or $S$ is a division ring.
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.
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
103830
Each element of a ring is either a unit or a nilpotent element iff the ring has a unique prime ideal
69373
Prove that $(0)$ is a radical ideal in $\mathbb{Z}/n\mathbb{Z}$ iff $n$ is square free
119067
Basic Ring Theory Question involving the unit element
24203
Finding Zero Divisors and Units of Rings $\mathbb{Z}_n \times \mathbb{Z}_m$
19838
Prove that in general the set of the right zero divisors is different from the set of left zero divisors in a ring R
138309
Ring whose all ideals are double-sided is commutative?
79957
If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors
15320
Morita equivalence and right and left ideals of a Ring
70320
Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?
43990
R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
69012
How to directly prove every element in a ring with exactly three ideals is either a unit or a zero divisor?
23641
Quotient ring of field is zero or itself.
65224
Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?
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.
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?