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?
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 $a,b \in R$.
41629
Structure of ideals in the product of two rings
79957
If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors
116651
Show for any simple ring without identity, R, that R is a division ring.
130814
Quotient of semi-simple rings
94532
How to prove a commutative, with unit, Noetherian ring $A$ only has finitely many minimal prime ideals via the following step?
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
63338
Noncommutative failure of IBN: $\,R^n\cong R^m \Rightarrow n - m,$ ring $R$ with maximal ideal $I$ such that $R/I$ is not a division ring?
127030
Simple example of non-arithmetic ring (non-distributive ideal lattice)
34183
An equivalent condition with $\{0\}$ being the only nilpotent ideal
89702
Let $R$ be a ring such that for all $a,b$ in $R$, $(a^2-a)b=b(a^2-a)$. Then $R$ is commutative
79897
if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian
91294
order of quotient ring
65536
Finitely generated ideals in a Boolean ring are principal, why?
79535
Prove that $R$ is a commutative ring if $x^3=x$
4397
Finite rings without zero divisors are division rings.
48009
Left inverse implies right inverse in a finite ring