Property: (right/left) T-nilpotent radical

Definition: (right T-nilpotent radical) Has right T-nilpotent Jacobson radical. Recall a nonempty subset $S$ of $R$ is right T-nilpotent if, for every sequence $\{r_i\mid i\in\mathbb N\}\subseteq S$, there exists an $n$ such that $r_nr_{n-1}\cdots r_0=0$. (The sequence of products is reversed for left T-nilpotent.)

Reference(s):

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Metaproperties:

This property does not have the following metaproperties
  • passes to subrings (Counterexample: $R_{ 44 }$ is a subring of $R_{ 15 }$)
  • stable under products (Counterexample: $R_{ 179 }$)
  • forms an equational class (counterexample needed)
  • passes to localizations (Counterexample: $R_{ 69 }$ is a localization of $R_{ 1 }$)
Rings
left Name right
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\widehat{\mathbb Z}$: the profinite completion of the integers
ACCP ring with non-ACCP power-series
Atomic domain with nonatomic polynomial ring
Bergman's example showing that "compressible" is not Morita invariant
Bergman's ring with IBN
Bergman's ring without IBN
Camillo and Nielsen's McCoy ring
Cohn's non-IBN domain
Cohn's right-not-left free ideal ring
Cohn's Schreier domain that isn't GCD
Eventually constant sequences in $\mathbb Z$
Facchini's torch ring
Faith-Menal counterexample
Kerr's Goldie ring with non-Goldie matrix ring
Kerr's non-Goldie matrix ring
Left-not-right Noetherian domain
Mori but not Krull domain
Nagata's Noetherian infinite Krull dimension ring
Nielsen's semicommutative ring that isn't McCoy
Noetherian domain that is not N-1
non-$h$-local domain
Non-ACCP polynomial ring
Non-ACCP power-series
Non-atomic polynomial ring
Non-IC polynomial ring over an IC ring
Non-Ore matrix ring over an Ore base ring
Page's left-not-right FPF ring
Principal ideal domain that is not Nagata
Ram's Ore extension ring
Right-not-left ACC on annihilators triangular 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
$2$-adic integers: $\mathbb Z_2$
$\mathbb Q[[X]]$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb R[[x]]$
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\varinjlim T_{2^n}(\Bbb Q)$
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$k[[x,y]]/(x^2,xy)$
$T_\omega(\mathbb Q)$
10-adic numbers
2-dimensional uniserial domain
ACCP ring with non-ACCP polynomial ring
Akizuki's counterexample
Base ring for $R_{187}$
catenary, not universally catenary
Clark's uniserial ring
Custom Krull dimension valuation ring
DVR that is not N-2
Grams' atomic domain which doesn't satisfy ACCP
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
Henselization of $\Bbb Z_{(2)}$
Interval monoid ring
Left-not-right pseudo-Frobenius ring
Left-not-right uniserial domain
Local right-not-left Kasch ring
Magnus algebra on two generators over $\mathbb Q$
Nagata ring that not quasi-excellent
Nagata's normal ring that is not analytically normal
Noetherian ring that is not Grothendieck and not Nagata
Non lift/rad matrix ring over a lift/rad base ring
Non strongly π-regular matrix ring over a strongly π-regular ring
Nonlocal endomorphism ring of a uniserial module
Osofsky's Type I ring
Pseudo-Frobenius, not quasi-Frobenius ring
Puninski's triangular serial ring
ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$
Samuel's UFD having a non-UFD power series ring
Square of a torch ring
Strongly π-regular base ring
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
$\mathbb H[X]$
$\mathbb Q$: the field of rational numbers
$\mathbb Q(x)$: rational functions over the rational numbers
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb R$: the field of real numbers
$\mathbb R[X,Y,Z]/(X^2+Y^2+Z^2-1)$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x,y]/(x^2+y^2-1)$: 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
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty F_2$
$\varinjlim \mathbb Q^{2^n}$
$\varinjlim M_{2^n}(\mathbb Q)$
$^\ast \mathbb R$: the field of hyperreal numbers
$C([0,1])$, the ring of continuous real-valued functions on the unit interval
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_2[x,y]/(x,y)^2$
$F_p(x)$
$k[x,x^{-1};\sigma]$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
$M_2(\mathbb H[X])$
$M_n(\mathbb Q)$
$M_n(F_2)$
$RCFM_\omega(\mathbb Q)$
$T_2(F_2)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
$T_n(F_2)$
$T_n(F_q)$
2-truncated Witt vectors over $\Bbb F_2((t))$
Algebra of differential operators on the line (1st Weyl algebra)
Algebraic closure of $F_2$
Algebraic integers
Algebraically closed division ring
Base ring for $R_{197}$
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 primitive finite uniform dimension ring
Bergman's right-not-left primitive ring
Bergman's unit-regular ring
Chase's left-not-right semihereditary ring
Cohn's quadratic extension skew field.
Countably infinite boolean ring
Cozzens simple, left principal, right non-Noetherian domain
Cozzens' simple V-domain
Division algebra with no anti-automorphism
Division ring with an antihomomorphism but no involution
Domanov's prime, nonprimitive, von Neumann regular ring
Eventually constant sequences in $\mathbb Q$
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
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
Hurwitz quaternions
Kaplansky's right-not-left hereditary ring
Kasch not semilocal ring
Kolchin's simple V-domain
Kronecker algebra $\Gamma_2(\mathbb Z)$
Leavitt path algebra of an infinite bouquet of circles
Lipschitz quaternions
Malcev's nonembeddable domain
McCoy ring that is not Abelian
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
Nielsen's right UGP, not left UGP ring
Non-Artinian simple ring
Non-IC matrix ring over an IC ring
Non-symmetric $2$-primal ring
Noncoherent product of coherent rings
O'Meara's infinite matrix algebra
Osofsky's $32$ element ring
Perfect non-Artinian ring
Perfect ring that isn't semiprimary
prime, von Neumann regular, nonprimitive Leavitt path algebra
Progression free polynomial ring
Quasi-continuous ring that is not Ikeda-Nakayama
Rational quaternions
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 Noetherian triangular ring
Right-not-left nonsingular ring
Right-not-left simple injective ring
Ring of holomorphic functions on $\mathbb C$
Ring of integer valued polynomials over the rationals
Simple, Noetherian ring with zero divisors and trivial idempotents
Simple, non-Artinian, von Neumann regular ring
Small's Noetherian, non-Ore ring
Šter's clean ring with non-clean corner rings
Legend
  • = has the property
  • = does not have the property
  • = information not in database