Property: top simple Artinian

Definition: $R/J(R)$ simple Artinian


(No citations retrieved.)


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