Property: (right/left) ACC principal

Definition: (right ACC principal) Ascending chain condition on principal right ideals

Reference(s):

(No citations retrieved.)

Metaproperties:

This property does not have the following metaproperties
Rings
left Name right
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$C([0,1])$, the ring of continuous real-valued functions on the unit interval
Base ring for $R_{191}$
Base ring for $R_{197}$
Bass's right-not-left perfect ring
Bergman's example showing that "compressible" is not Morita invariant
Bergman's primitive finite uniform dimension ring
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 non-IBN domain
Cozzens simple, left principal, right non-Noetherian domain
Facchini's torch ring
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
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
Left-not-right uniserial domain
Local right-not-left Kasch ring
Magnus algebra on two generators over $\mathbb Q$
Malcev's nonembeddable domain
McGovern's commutative Zorn ring that isn't clean
Nielsen's semicommutative ring that isn't McCoy
Non π-regular matrix ring over a π-regular ring
non-$h$-local domain
Non-Artinian simple ring
Non-Ore matrix ring over an Ore base ring
Progression free polynomial ring
Pseudo-Frobenius, not quasi-Frobenius ring
Puninski's triangular serial ring
Ram's Ore extension ring
Right-not-left simple injective ring
Semicommutative $R$ such that $R[x]$ is not semicommutative
Shepherdson's domain that is not stably finite
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 $\mathbb Q$
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb Z+x\mathbb Q[x]$
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\prod_{i=1}^\infty F_2$
$\varinjlim T_{2^n}(\Bbb Q)$
$\varinjlim \mathbb Q^{2^n}$
$\varinjlim M_{2^n}(\mathbb Q)$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$RCFM_\omega(\mathbb Q)$
$T_\omega(\mathbb Q)$
2-dimensional uniserial domain
Algebraic integers
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
Clark's uniserial ring
Cohn's Schreier domain that isn't GCD
Countably infinite boolean ring
Custom Krull dimension valuation ring
Domanov's prime, nonprimitive, von Neumann regular 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
Hochster's connected, nondomain, locally-domain ring
Interval monoid ring
Kaplansky's right-not-left hereditary ring
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
prime, von Neumann regular, nonprimitive Leavitt path algebra
reduced $I_0$ ring that is not exchange
reduced exchange ring which is not semiregular
Ring of holomorphic functions on $\mathbb C$
Simple, non-Artinian, von Neumann regular 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
$\mathbb H[X]$
$\mathbb Q$: the field of rational numbers
$\mathbb Q(x)$: rational functions over the rational numbers
$\mathbb Q[[X]]$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb R$: the field of real numbers
$\mathbb R[[x]]$
$\mathbb R[X,Y,Z]/(X^2+Y^2+Z^2-1)$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x,y]$ 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
$\mathbb Z/(2)$
$\mathbb Z/(n)$, $n$ 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
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_2[x,y]/(x,y)^2$
$F_p(x)$
$k[[x,y]]/(x^2,xy)$
$k[x,x^{-1};\sigma]$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
$M_2(\mathbb H[X])$
$M_n(\mathbb Q)$
$M_n(F_2)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
$T_n(F_2)$
$T_n(F_q)$
10-adic numbers
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 $F_2$
Base ring for $R_{187}$
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
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-IC matrix ring over an IC ring
Non-IC polynomial ring over an IC ring
Non-symmetric $2$-primal ring
Osofsky's $32$ element ring
Perfect non-Artinian ring
Perfect ring that isn't semiprimary
Principal ideal domain that is not Nagata
Quasi-continuous ring that is not Ikeda-Nakayama
Rational quaternions
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 $\mathbb C^n$, $n>1$
Ring of integer valued polynomials over the rationals
Samuel's UFD having a non-UFD power series ring
Simple, Noetherian ring with zero divisors and trivial idempotents
Small's Noetherian, non-Ore ring
Varadarajan's left-not-right coHopfian ring
Legend
  • = has the property
  • = does not have the property
  • = information not in database