Property: Zorn

Definition: $R$ is $I_0$ and $J(R)$ is a nil ideal

Reference(s):

(No citations retrieved.)

Metaproperties:

This property does not have the following metaproperties
Rings
Name
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$RCFM_\omega(\mathbb Q)$
$T_\omega(\mathbb Q)$
Bergman's example showing that "compressible" is not Morita invariant
Bergman's right-not-left primitive ring
Bergman's ring with IBN
Bergman's ring without IBN
Camillo and Nielsen's McCoy ring
Cohn's right-not-left free ideal ring
Cohn's Schreier domain that isn't GCD
Eventually constant sequences in $\mathbb Z$
Faith-Menal counterexample
Grams' atomic domain which doesn't satisfy ACCP
Hurwitz quaternions
Kasch not semilocal ring
Kerr's Goldie ring with non-Goldie matrix ring
Leavitt path algebra of an infinite bouquet of circles
Left-not-right Noetherian domain
Left-not-right pseudo-Frobenius ring
Lipschitz quaternions
Mori but not Krull domain
Nielsen's semicommutative ring that isn't McCoy
Noetherian domain that is not N-1
Non-Artinian simple ring
O'Meara's infinite matrix algebra
Page's left-not-right FPF ring
Puninski's triangular serial ring
Right-not-left ACC on annihilators triangular ring
Semicommutative $R$ such that $R[x]$ is not semicommutative
Small's right hereditary, not-left semihereditary ring
Square of a torch ring
Šter's counterexample showing "clean" is not Morita invariant
$2$-adic integers: $\mathbb Z_2$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb R[[x]]$
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
$\mathbb R[x_1, x_2,x_3,\ldots]$
$\mathbb Z$: the ring of integers
$\mathbb Z+x\mathbb Q[x]$
$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
$\mathbb Z[\sqrt{-5}]$
$\mathbb Z[i]$: the Gaussian integers
$\mathbb Z[x]$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$C([0,1])$, the ring of continuous real-valued functions on the unit interval
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$k[[x,y]]/(x^2,xy)$
$k[x,x^{-1};\sigma]$
$k[x,y,z]/(xz,yz)$
$k[x,y]/(x^2, xy)$
$k[x,y]/(x^2-y^3)$
$k[x,y]_{(x,y)}/(x^2-y^3)$
10-adic numbers
2-dimensional uniserial domain
Akizuki's counterexample
Algebra of differential operators on the line (1st Weyl algebra)
Algebraic integers
Bergman's primitive finite uniform dimension ring
catenary, not universally catenary
Clark's uniserial ring
Cohn's non-IBN domain
Cozzens simple, left principal, right non-Noetherian domain
Cozzens' simple V-domain
Custom Krull dimension valuation ring
DVR that is not N-2
Facchini's torch ring
Finitely cogenerated, not semilocal ring.
Henselization of $\Bbb Z_{(2)}$
Hochster's connected, nondomain, locally-domain ring
Kolchin's simple V-domain
Left-not-right uniserial domain
Local right-not-left Kasch ring
Malcev's nonembeddable domain
Nagata ring that not quasi-excellent
Nagata's Noetherian infinite Krull dimension ring
Nagata's normal ring that is not analytically normal
Nielsen's right UGP, not left UGP ring
Noetherian ring that is not Grothendieck and not Nagata
non-$h$-local domain
Non-symmetric $2$-primal ring
Nonlocal endomorphism ring of a uniserial module
Osofsky's Type I ring
Progression free polynomial ring
Pseudo-Frobenius, not quasi-Frobenius ring
Ram's Ore extension ring
Right-not-left Noetherian triangular ring
Right-not-left nonsingular ring
ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$
Ring of holomorphic functions on $\mathbb C$
Samuel's UFD having a non-UFD power series ring
Shepherdson's domain that is not stably finite
Simple, connected, Noetherian ring with zero divisors
Trivial extension torch ring
Varadarajan's left-not-right coHopfian ring
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb C$: the field of complex numbers
$\mathbb H$: Hamilton's quaternions
$\mathbb Q$: the field of rational numbers
$\mathbb Q(x)$: rational functions over the rational numbers
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb R$: the field of real numbers
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x]/(x^2)$
$\mathbb Z/(2)$
$\mathbb Z/(n)$, $n$ divisible by two primes and a square
$\mathbb Z/(n)$, $n$ squarefree and not prime.
$\mathbb Z/(p)$, $p$ an odd prime
$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\prod_{i=1}^\infty F_2$
$^\ast \mathbb R$: the field of hyperreal numbers
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_2[x,y]/(x,y)^2$
$F_p(x)$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$M_n(\mathbb Q)$
$M_n(F_2)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
$T_n(F_2)$
$T_n(F_q)$
Algebraic closure of $F_2$
Basic ring of Nakayama's QF ring
Bass's right-not-left perfect ring
Berberian's incompressible Baer ring
Bergman's exchange ring that isn't clean
Bergman's non-unit-regular subring
Bergman's unit-regular ring
Chase's left-not-right semihereditary ring
Countably infinite boolean ring
Division algebra with no anti-automorphism
Division ring with an antihomomorphism but no involution
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
Full linear ring of a countable dimensional right vector space
Goodearl's simple self-injective operator algebra
Goodearl's simple self-injective von Neumann regular ring
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
Interval monoid ring
Kaplansky's right-not-left hereditary ring
McGovern's commutative Zorn ring that isn't clean
Michler & Villamayor's right-not-left V ring
Nakayama's quasi-Frobenius ring that isn't Frobenius
Osofsky's $32$ element ring
Perfect non-Artinian ring
Perfect ring that isn't semiprimary
Quasi-continuous ring that is not Ikeda-Nakayama
reduced $I_0$ ring that is not exchange
reduced exchange ring which is not semiregular
Reversible non-symmetric ring
Right-not-left Artinian triangular ring
Right-not-left coherent ring
Right-not-left Kasch ring
Right-not-left simple injective ring
Simple, non-Artinian, von Neumann regular ring
Šter's clean ring with non-clean corner rings
Legend
  • = has the property
  • = does not have the property
  • = information not in database