Example of a ring $R$ in which $a^2=0 \implies a \in J(R)$ but $N(R) \nsubseteq J(R)$.
18003
Standard example where Jacobson radical not equal nilradical
91962
In a ring $a*a=0$ then $a+a=0$
84751
How can it be possible to localize a ring at a set with zero divisors?
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
71977
Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive
154613
Showing a set of fractions is a ring
115920
Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
111679
Example of a ring without unity that has a subring with unity?
86816
Example of two prime ideals whose intersection of the squares not equal to the square of the intersection
76749
Example of a finite non-commutative ring without a unity
95408
Give an example to show that a factor ring of a ring with divisors of 0 may be an integral domain
34183
An equivalent condition with $\{0\}$ being the only nilpotent ideal
114254
Zero divisors in $A[x_1,x_2,\dots,x_r]$
148616
When is a ring homomorphism $A \to A$ surjective but not injective.
148788
Polynomial ring ideal. Show that $(f) \cap R_N = (0) \iff f \notin R_N$
68807
How to show that the ring $S/A$ has no zero divisors? (Hungerford, Algebra, Problem 12, Chapter III, Section 2)
28327
(Simple) Examples on Non-Commutative Rings