Property: (right/left) perfect

Definition: (right perfect) $R$ is semilocal and $J(R)$ is right T-nilpotent

Reference(s):

  • A. A. Tuganbaev. Semidistributive modules and rings. (2012) @ Chapter 6
  • F. W. Anderson and K. R. Fuller. Rings and categories of modules. (2012) @ Chapter 7, Section 28
  • T.-Y. Lam. A first course in noncommutative rings. (2013) @ Chapter 8

Metaproperties:

This property has the following metaproperties
  • passes to $eRe$ for any idempotent $e$
  • passes to $eRe$ for any full idempotent $e$
  • Morita invariant
  • passes to matrix rings
This property does not have the following metaproperties
Rings
left Name right
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$T_\omega(\mathbb Q)$
Camillo and Nielsen's McCoy ring
Eventually constant sequences in $\mathbb Z$
Kerr's Goldie ring with non-Goldie matrix ring
Nielsen's semicommutative ring that isn't McCoy
Semicommutative $R$ such that $R[x]$ is not semicommutative
Small's right hereditary, not-left semihereditary ring
$2$-adic integers: $\mathbb Z_2$
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb R[[x]]$
$\mathbb R[x,y]$ 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
$\mathbb Z+x\mathbb Q[x]$
$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
$\mathbb Z[\sqrt{-5}]$
$\mathbb Z[i]$: 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 $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\prod_{i=1}^\infty F_2$
$\widehat{\mathbb Z}$: 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 $0$
$k[[x,y]]/(x^2,xy)$
$k[x,x^{-1};\sigma]$
$k[x,y,z]/(xz,yz)$
$k[x,y]/(x^2, xy)$
$k[x,y]/(x^2-y^3)$
$k[x,y]_{(x,y)}/(x^2-y^3)$
$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$RCFM_\omega(\mathbb Q)$
10-adic numbers
2-dimensional uniserial domain
Akizuki's counterexample
Algebra of differential operators on the line (1st Weyl algebra)
Algebraic integers
Bergman's example showing that "compressible" is not Morita invariant
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
catenary, not universally catenary
Chase's left-not-right semihereditary ring
Clark's uniserial ring
Cohn's non-IBN domain
Cohn's right-not-left free ideal ring
Cohn's Schreier domain that isn't GCD
Countably infinite boolean ring
Cozzens simple, left principal, right non-Noetherian domain
Cozzens' simple V-domain
Custom Krull dimension valuation ring
DVR that is not N-2
Facchini's torch ring
Faith-Menal counterexample
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 $\bigwedge (V)$, $\dim(V)=\aleph_0$
Henselization of $\Bbb Z_{(2)}$
Hochster's connected, nondomain, locally-domain ring
Hurwitz quaternions
Interval monoid 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
Left-not-right Noetherian domain
Left-not-right pseudo-Frobenius ring
Left-not-right uniserial domain
Lipschitz quaternions
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
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
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-$h$-local domain
Non-Artinian simple ring
Non-symmetric $2$-primal 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
Progression free polynomial ring
Pseudo-Frobenius, not quasi-Frobenius ring
Puninski's triangular serial ring
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
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, connected, Noetherian ring with zero divisors
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
Šter's counterexample showing "clean" is not Morita invariant
$\mathbb C$: 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
$\mathbb R$: the field of real numbers
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x]/(x^2)$
$\mathbb Z/(2)$
$\mathbb Z/(n)$, $n$ 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, $k>1$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$^\ast \mathbb R$: the field of hyperreal numbers
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_2[x,y]/(x,y)^2$
$F_p(x)$
$k[x;\sigma]/(x^2)$ (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 $\mathbb Q$
$T_n(F_2)$
$T_n(F_q)$
Algebraic closure of $F_2$
Basic ring of Nakayama's QF ring
Bass's right-not-left perfect ring
Berberian's incompressible Baer ring
Division algebra with no anti-automorphism
Division ring with an antihomomorphism but no involution
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
Nakayama's quasi-Frobenius ring that isn't Frobenius
Osofsky's $32$ element ring
Perfect non-Artinian ring
Perfect ring that isn't semiprimary
Quasi-continuous ring that is not Ikeda-Nakayama
Reversible non-symmetric ring
Right-not-left Artinian triangular ring
Right-not-left coherent ring
Right-not-left Kasch ring
Right-not-left simple injective ring
Legend
  • = has the property
  • = does not have the property
  • = information not in database