Property: orthogonally finite

Definition: families of orthogonal idempotents are all finite


(No citations retrieved.)


This property has the following metaproperties
  • passes to subrings
  • passes to the center
This property does not have the following metaproperties
Bergman's right-not-left primitive ring
Bergman's ring with IBN
Bergman's ring without IBN
Camillo and Nielsen's McCoy ring
Eventually constant sequences in $\mathbb Z$
Kaplansky's right-not-left hereditary ring
Kasch not semilocal ring
Leavitt path algebra of an infinite bouquet of circles
McGovern's commutative Zorn ring that isn't clean
Nielsen's semicommutative ring that isn't McCoy
Non-Artinian simple ring
Puninski's triangular serial ring
reduced $I_0$ ring that is not exchange
reduced exchange ring which is not semiregular
Semicommutative $R$ such that $R[x]$ is not semicommutative
Small's right hereditary, not-left semihereditary ring
Square of a torch ring
Šter's counterexample showing "clean" is not Morita invariant
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q\langle x,y \rangle/(xy-1)$: 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 unit-regular ring
Chase's left-not-right semihereditary ring
Countably infinite boolean 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
Michler & Villamayor's right-not-left V ring
O'Meara's infinite matrix algebra
Page's left-not-right FPF 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
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x]$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb R$: 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
$C([0,1])$, the ring of continuous real-valued functions on the unit interval
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$F_2[\mathcal Q_8]$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
$M_n(\mathbb Q)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
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
Basic ring of Nakayama's QF ring
Bass's right-not-left perfect ring
Berberian's incompressible Baer ring
Bergman's example showing that "compressible" is not Morita invariant
Bergman's primitive finite uniform dimension ring
catenary, not universally catenary
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
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
Facchini's torch ring
Faith-Menal counterexample
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
Finitely cogenerated, not semilocal 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
Kerr's Goldie ring with non-Goldie matrix ring
Kolchin's simple V-domain
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
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-$h$-local domain
Non-symmetric $2$-primal ring
Nonlocal endomorphism ring of a uniserial module
Osofsky's $32$ element ring
Osofsky's Type I 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
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
Right-not-left simple injective 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
Trivial extension torch ring
Varadarajan's left-not-right coHopfian ring
  • = has the property
  • = does not have the property
  • = information not in database