Description: Consider the semigroup $S=\{a,b\}$ with $a^2=ab=a$,$b^2=ba=b$. Make the semigroup ring $T=F_2[S]$. Make the Dorroh extension $T'=\mathbb Z(+)T$. $T'$ is the ring.

Notes:

Keywords Dorroh extension

Reference(s):

- Greg Marks, Reversible And Symmetric Rings, Journal Of Pure And Applied Algebra Volume 174, Issue 3, Pp 311–318, (2002). Example 2 P 313

This ring has the following properties:

2-primal
ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
coherent (left)
coherent (right)
DCC annihilator (left)
DCC annihilator (right)
Dedekind finite
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
Goldie (left)
Goldie (right)
IBN
lift/rad
NI (nilpotents form an ideal)
nil radical
nilpotent radical
Noetherian (left)
Noetherian (right)
orthogonally finite
stably finite
T-nilpotent radical (left)
T-nilpotent radical (right)

The ring lacks the following properties:

$\pi$-regular
$I_0$
Abelian
Artinian (left)
Artinian (right)
Bezout domain (left)
Bezout domain (right)
clean
cogenerator ring (left)
cogenerator ring (right)
commutative
continuous (left)
continuous (right)
distributive (left)
distributive (right)
division ring
domain
dual (left)
dual (right)
duo (left)
duo (right)
exchange
FI-injective (left)
FI-injective (right)
finite
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Kasch (left)
Kasch (right)
local
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principally injective (left)
principally injective (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
reduced
reversible
self-injective (left)
self-injective (right)
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiprimary
semiprime
semiprimitive
semiregular
semisimple
serial (left)
serial (right)
simple
simple Artinian
strongly $\pi$-regular
strongly connected
strongly regular
symmetric
top regular
top simple Artinian
uniform (left)
uniform (right)
unit regular
V ring (left)
V ring (right)
valuation ring (left)
valuation ring (right)
von Neumann regular
Zorn

We don't know if the ring has or lacks the following properties:

Baer
Bezout (left)
Bezout (right)
cohopfian (left)
cohopfian (right)
connected
CS (left)
CS (right)
essential socle (left)
essential socle (right)
finitely cogenerated (left)
finitely cogenerated (right)
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
hereditary (left)
hereditary (right)
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
nonsingular (left)
nonsingular (right)
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
polynomial identity
principal ideal ring (left)
principal ideal ring (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
Rickart (left)
Rickart (right)
semihereditary (left)
semihereditary (right)
simple socle (left)
simple socle (right)
simple-injective (left)
simple-injective (right)
stable range 1
top simple
weakly clean