Description: Let the interval $I=[0,1]$ in the real numbers be a monoid under addition, where $a+b:=0$ if $a+b >1$. The ring is the monoid ring of $I$ over a field $F$.

Notes: $J(R)$ is idempotent and nil. Krull dimension $0$.

Keywords semigroup ring

Reference(s):

- Hajarnavis, C. R. ; Norton, N. C., On Dual Rings And Their Modules, Journal Of Algebra 93 P253-266, (1985). Example 6.2 P 265-266
- N. C. Norton, Generalizations Of The Theory Of Quasi-Frobenius Rings, Doctoral Dissertation For University Of Warwick, (1975). Example 3.2.2 P 112

This ring has the following properties:

$\pi$-regular
$I_0$
2-primal
Abelian
Bezout (left)
Bezout (right)
clean
cohopfian (left)
cohopfian (right)
commutative
connected
continuous (left)
continuous (right)
CS (left)
CS (right)
Dedekind finite
distributive (left)
distributive (right)
dual (left)
dual (right)
duo (left)
duo (right)
exchange
FI-injective (left)
FI-injective (right)
finite uniform dimension (left)
finite uniform dimension (right)
finitely generated socle (left)
finitely generated socle (right)
IBN
Ikeda-Nakayama (left)
Ikeda-Nakayama (right)
Kasch (left)
Kasch (right)
lift/rad
local
NI (nilpotents form an ideal)
nil radical
nonzero socle (left)
nonzero socle (right)
Ore ring (left)
Ore ring (right)
orthogonally finite
polynomial identity
principally injective (left)
principally injective (right)
quasi-continuous (left)
quasi-continuous (right)
quasi-duo (left)
quasi-duo (right)
reversible
semicommutative (SI condition, zero-insertive)
semilocal
semiperfect
semiregular
serial (left)
serial (right)
simple-injective (left)
simple-injective (right)
stable range 1
stably finite
strongly $\pi$-regular
strongly connected
symmetric
top regular
top simple
top simple Artinian
uniform (left)
uniform (right)
valuation ring (left)
valuation ring (right)
weakly clean
Zorn

The ring lacks the following properties:

ACC annihilator (left)
ACC annihilator (right)
ACC principal (left)
ACC principal (right)
Artinian (left)
Artinian (right)
Baer
Bezout domain (left)
Bezout domain (right)
cogenerator ring (left)
cogenerator ring (right)
DCC annihilator (left)
DCC annihilator (right)
division ring
domain
finite
finitely pseudo-Frobenius (left)
finitely pseudo-Frobenius (right)
free ideal ring (left)
free ideal ring (right)
Frobenius
fully prime
fully semiprime
Goldie (left)
Goldie (right)
hereditary (left)
hereditary (right)
nilpotent radical
Noetherian (left)
Noetherian (right)
nonsingular (left)
nonsingular (right)
Ore domain (left)
Ore domain (right)
perfect (left)
perfect (right)
primary
prime
primitive (left)
primitive (right)
principal ideal domain (left)
principal ideal domain (right)
principal ideal ring (left)
principal ideal ring (right)
pseudo-Frobenius (left)
pseudo-Frobenius (right)
quasi-Frobenius
reduced
Rickart (left)
Rickart (right)
self-injective (left)
self-injective (right)
semi free ideal ring
semihereditary (left)
semihereditary (right)
semiprimary
semiprime
semiprimitive
semisimple
simple
simple Artinian
strongly regular
T-nilpotent radical (left)
T-nilpotent radical (right)
unit regular
V ring (left)
V ring (right)
von Neumann regular

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