Property: (right/left) DCC annihilator

Definition: (right DCC annihilator) Descending chain condition on right annihilators=Ascending chain condition on left annihilators

Reference(s):

(No citations retrieved.)

Metaproperties:

This property has the following metaproperties
  • passes to subrings
  • passes to the center
This property does not have the following metaproperties
  • stable under products (Counterexample: $R_{ 57 }$)
  • forms an equational class (counterexample needed)
  • passes to quotient rings (Counterexample: $R_{ 119 }$ is a homomorphic image of $R_{ 165 }$)
Rings
left Name right
$C([0,1])$, 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 $0$
Bass's right-not-left perfect ring
Bergman's right-not-left primitive ring
Bergman's ring with IBN
Bergman's ring without IBN
Camillo and Nielsen's McCoy ring
Facchini's torch ring
Finitely cogenerated, not semilocal ring
Kasch not semilocal ring
Leavitt path algebra of an infinite bouquet of circles
Local right-not-left Kasch ring
McGovern's commutative Zorn ring that isn't clean
Nielsen's semicommutative ring that isn't McCoy
Non-Artinian simple ring
Nonlocal endomorphism ring of a uniserial module
Perfect non-Artinian ring
Perfect ring that isn't semiprimary
Puninski's triangular serial ring
Right-not-left simple injective ring
Semicommutative $R$ such that $R[x]$ is not semicommutative
Small's right hereditary, not-left semihereditary ring
Square of a torch ring
Trivial extension torch ring
Šter's counterexample showing "clean" is not Morita invariant
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\prod_{i=1}^\infty F_2$
$\varinjlim T_{2^n}(\Bbb Q)$
$\varinjlim \mathbb Q^{2^n}$
$\varinjlim M_{2^n}(\mathbb Q)$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$RCFM_\omega(\mathbb Q)$
$T_\omega(\mathbb Q)$
Bergman's exchange ring that isn't clean
Bergman's non-unit-regular subring
Bergman's primitive finite uniform dimension ring
Bergman's unit-regular ring
Chase's left-not-right semihereditary ring
Clark's uniserial ring
Countably infinite boolean ring
Eventually constant sequences in $\mathbb Z$
Faith-Menal counterexample
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 $\bigwedge (V)$, $\dim(V)=\aleph_0$
Hochster's connected, nondomain, locally-domain ring
Interval monoid ring
Kaplansky's right-not-left hereditary ring
Left-not-right pseudo-Frobenius ring
Michler & Villamayor's right-not-left V ring
O'Meara's infinite matrix algebra
Page's left-not-right FPF ring
Progression free polynomial ring
Pseudo-Frobenius, not quasi-Frobenius ring
Quasi-continuous ring that is not Ikeda-Nakayama
Ram's Ore extension ring
reduced $I_0$ ring that is not exchange
reduced exchange ring which is not semiregular
Right-not-left ACC on annihilators triangular ring
Simple, non-Artinian, von Neumann regular 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
$\mathbb Q[[X]]$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb R$: the field of real numbers
$\mathbb R[[x]]$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
$\mathbb R[x]/(x^2)$
$\mathbb R[x_1, x_2,x_3,\ldots]$
$\mathbb Z$: the ring of integers
$\mathbb Z+x\mathbb Q[x]$
$\mathbb Z/(2)$
$\mathbb Z/(n)$, $n$ 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, $k>1$
$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
$\mathbb Z[\sqrt{-5}]$
$\mathbb Z[i]$: the Gaussian integers
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$^\ast \mathbb R$: the field of hyperreal numbers
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_2[x,y]/(x,y)^2$
$F_p(x)$
$k[[x,y]]/(x^2,xy)$
$k[x,x^{-1};\sigma]$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
$M_n(\mathbb Q)$
$M_n(F_2)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
$T_n(F_2)$
$T_n(F_q)$
10-adic numbers
2-dimensional uniserial domain
2-truncated Witt vectors over $\Bbb F_2((t))$
Akizuki's counterexample
Algebra of differential operators on the line (1st Weyl algebra)
Algebraic closure of $F_2$
Algebraic integers
Base ring for $R_{187}$
Basic ring of Nakayama's QF ring
Berberian's incompressible Baer ring
Bergman's example showing that "compressible" is not Morita invariant
catenary, not universally catenary
Cohn's non-IBN domain
Cohn's right-not-left free ideal ring
Cohn's Schreier domain that isn't GCD
Cozzens simple, left principal, right non-Noetherian domain
Cozzens' simple V-domain
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
Grams' atomic domain which doesn't satisfy ACCP
Henselization of $\Bbb Z_{(2)}$
Hurwitz quaternions
Kerr's Goldie ring with non-Goldie matrix ring
Kolchin's simple V-domain
Left-not-right Noetherian domain
Left-not-right uniserial domain
Lipschitz quaternions
Malcev's nonembeddable domain
McCoy ring that is not Abelian
Mori but not Krull domain
Nagata ring that not quasi-excellent
Nagata's Noetherian infinite Krull dimension ring
Nagata's normal ring that is not analytically normal
Nakayama's quasi-Frobenius ring that isn't Frobenius
Nielsen's right UGP, not left UGP ring
Noetherian domain that is not N-1
Noetherian ring that is not Grothendieck and not Nagata
Non lift/rad matrix ring over a lift/rad base ring
non-$h$-local domain
Non-symmetric $2$-primal ring
Osofsky's $32$ element ring
Osofsky's Type I ring
Rational quaternions
Reversible non-symmetric ring
Right-not-left Artinian triangular ring
Right-not-left coherent ring
Right-not-left Kasch 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, Noetherian ring with zero divisors and trivial idempotents
Varadarajan's left-not-right coHopfian ring
Legend
  • = has the property
  • = does not have the property
  • = information not in database