Ring detail


Name: Interval monoid ring

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):

  • C. Hajarnavis and N. Norton. On dual rings and their modules. (1985) @ Example 6.2, pp 265-266
  • N. C. Norton. Generalizations of the theory of quasi-frobenius rings. (1975) @ Example 3.2.2, p 112



Known Properties
Name
$I_0$
$\pi$-regular
2-primal
ACC annihilator
ACC principal
Abelian
Archimedean field
Artinian
Baer
Bezout
Bezout domain
CS
Cohen-Macaulay
DCC annihilator
Dedekind domain
Dedekind finite
Euclidean domain
Euclidean field
FI-injective
Frobenius
GCD domain
Goldie
Gorenstein
Grothendieck
IBN
Ikeda-Nakayama
J-0
J-1
J-2
Jacobson (Hilbert)
Japanese (N-2)
Kasch
Krull domain
Mori domain
N-1
NI (nilpotents form an ideal)
Nagata
Noetherian
Ore domain
Ore ring
Prufer domain
Pythagorean field
Rickart
Schreier domain
T-nilpotent radical
V ring
Zorn
algebraically closed field
analytically normal
analytically unramified
atomic domain
catenary
characteristic 0 field
clean
cogenerator ring
coherent
cohopfian
commutative
complete local
compressible
connected
continuous
distributive
division ring
domain
dual
duo
essential socle
excellent
exchange
field
finite
finite uniform dimension
finitely cogenerated
finitely generated socle
finitely pseudo-Frobenius
free ideal ring
fully prime
fully semiprime
hereditary
lift/rad
local
local complete intersection
nil radical
nilpotent radical
nonsingular
nonzero socle
normal
normal domain
ordered field
orthogonally finite
perfect
perfect field
polynomial identity
primary
prime
primitive
principal ideal domain
principal ideal ring
principally injective
pseudo-Frobenius
quadratically closed field
quasi-Frobenius
quasi-continuous
quasi-duo
quasi-excellent
rad-nil
reduced
regular
regular local
reversible
self-injective
semi free ideal ring
semicommutative (SI condition, zero-insertive)
semihereditary
semilocal
semiperfect
semiprimary
semiprime
semiprimitive
semiregular
semisimple
serial
simple
simple Artinian
simple socle
simple-injective
stable range 1
stably finite
strongly $\pi$-regular
strongly connected
strongly regular
symmetric
top regular
top simple
top simple Artinian
uniform
unique factorization domain
unit regular
universally Japanese
universally catenary
valuation ring
von Neumann regular
weakly clean