Property: (right/left) Noetherian

Definition: (right Noetherian) Ascending chain condition on right ideals

Reference(s):

  • J. C. McConnell, J. C. Robson, and L. W. Small. Noncommutative {N}oetherian rings. (2001) @ .
  • K. R. Goodearl and R. B. W. Jr. An introduction to noncommutative Noetherian rings. (2004) @ .
  • N. Jacobson. Basic algebra II. (2012) @ Section 7.10

Metaproperties:

This property has the following metaproperties
  • stable under finite products
  • Morita invariant
  • passes to localizations
  • passes to matrix rings
  • passes to $eRe$ for any full idempotent $e$
  • passes to quotient rings
This property does not have the following metaproperties
  • stable under products (Counterexample: $R_{ 57 }$)
  • forms an equational class (counterexample needed)
  • passes to subrings (Counterexample: $R_{ 36 }$ is a subring of $R_{ 7 }$)
Rings
left Name right
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q\langle a,b\rangle/(a^2)$
$T_\omega(\mathbb Q)$
Bergman's example showing that "compressible" is not Morita invariant
Bergman's right-not-left primitive ring
Camillo and Nielsen's McCoy ring
Facchini's torch ring
Leavitt path algebra of an infinite bouquet of circles
Mori but not Krull domain
Nielsen's semicommutative ring that isn't McCoy
non-$h$-local domain
Non-Artinian simple ring
Puninski's triangular serial ring
Semicommutative $R$ such that $R[x]$ is not semicommutative
Šter's counterexample showing "clean" is not Morita invariant
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[\mathbb Q]$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb R[x_1, x_2,x_3,\ldots]$
$\mathbb Z+x\mathbb Q[x]$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\prod_{i=1}^\infty F_2$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$C([0,1])$, 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 $0$
$k[x;\sigma]/(x^2)$ (not right Artinian)
$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$RCFM_\omega(\mathbb Q)$
2-dimensional uniserial domain
Algebraic integers
Bass's right-not-left perfect ring
Bergman's exchange ring that isn't clean
Bergman's non-unit-regular subring
Bergman's primitive finite uniform dimension ring
Bergman's ring with IBN
Bergman's ring without IBN
Bergman's unit-regular ring
Chase's left-not-right semihereditary ring
Clark's uniserial ring
Cohn's non-IBN domain
Cohn's Schreier domain that isn't GCD
Countably infinite boolean ring
Cozzens simple, left principal, right non-Noetherian domain
Custom Krull dimension valuation ring
Eventually constant sequences in $\mathbb Z$
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
Grams' atomic domain which doesn't satisfy ACCP
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
Kasch not semilocal ring
Kerr's Goldie ring with non-Goldie matrix ring
Left-not-right Noetherian domain
Left-not-right pseudo-Frobenius ring
Left-not-right uniserial domain
Local right-not-left Kasch ring
Malcev's nonembeddable domain
McGovern's commutative Zorn ring that isn't clean
Michler & Villamayor's right-not-left V ring
Nonlocal endomorphism ring of a uniserial module
O'Meara's infinite matrix algebra
Osofsky's Type I ring
Page's left-not-right FPF ring
Perfect non-Artinian ring
Perfect ring that isn't semiprimary
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 simple injective ring
Ring of holomorphic functions on $\mathbb C$
Shepherdson's domain that is not stably finite
Simple, non-Artinian, von Neumann regular ring
Small's right hereditary, not-left semihereditary ring
Square of a torch ring
Trivial extension torch 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
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb R$: 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
$\mathbb R[x]/(x^2)$
$\mathbb Z$: the ring of integers
$\mathbb Z/(2)$
$\mathbb Z/(n)$, $n$ 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
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z_S$, 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
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_2[x,y]/(x,y)^2$
$F_p(x)$
$k[[x,y]]/(x^2,xy)$
$k[x,x^{-1};\sigma]$
$k[x,y,z]/(xz,yz)$
$k[x,y]/(x^2, xy)$
$k[x,y]/(x^2-y^3)$
$k[x,y]_{(x,y)}/(x^2-y^3)$
$k[x;\sigma]/(x^2)$ (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
Akizuki's counterexample
Algebra of differential operators on the line (1st Weyl algebra)
Algebraic closure of $F_2$
Basic ring of Nakayama's QF ring
Berberian's incompressible Baer ring
catenary, not universally catenary
Cohn's right-not-left free ideal ring
Cozzens' simple V-domain
Division algebra with no anti-automorphism
Division ring with an antihomomorphism but no involution
DVR that is not N-2
Faith-Menal counterexample
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
Henselization of $\Bbb Z_{(2)}$
Hurwitz quaternions
Kolchin's simple V-domain
Lipschitz quaternions
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-symmetric $2$-primal ring
Osofsky's $32$ element ring
Reversible non-symmetric ring
Right-not-left ACC on annihilators triangular 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 $\mathbb C^n$, $n>1$
Samuel's UFD having a non-UFD power series ring
Simple, connected, Noetherian ring with zero divisors
Varadarajan's left-not-right coHopfian ring
Legend
  • = has the property
  • = does not have the property
  • = information not in database