# Property: (right/left) quasi-continuous

Definition: (right quasi-continuous) $R$ is right CS, and if $e,f$ are idempotents with $eR\cap fR=\{0\}$, then $eR\oplus fR$ is a summand of $R$.

### Reference(s):

(No citations retrieved.)

### Metaproperties:

This property does not have the following metaproperties
Rings
left Name right
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\prod_{i=0}^\infty \mathbb Q$
$\varinjlim T_{2^n}(\Bbb Q)$
$\varinjlim \mathbb Q^{2^n}$
$\varinjlim M_{2^n}(\mathbb Q)$
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$k[[x,y]]/(x^2,xy)$
$RCFM_\omega(\mathbb Q)$
$T_\omega(\mathbb Q)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
$T_n(F_q)$
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 ring with IBN
Bergman's ring without IBN
Bergman's unit-regular ring
Camillo and Nielsen's McCoy ring
Chase's left-not-right semihereditary ring
Cohn's right-not-left free ideal ring
Countably infinite boolean ring
Eventually constant sequences in $\mathbb Z$
Facchini's torch ring
Faith-Menal counterexample
Full linear ring of a countable dimensional right vector space
Kaplansky's right-not-left hereditary ring
Kerr's Goldie ring with non-Goldie matrix ring
Leavitt path algebra of an infinite bouquet of circles
Local right-not-left Kasch ring
McGovern's commutative Zorn ring that isn't clean
Michler & Villamayor's right-not-left V ring
Nagata's normal ring that is not analytically normal
Nielsen's right UGP, not left UGP ring
Nielsen's semicommutative ring that isn't McCoy
Non-Artinian simple ring
Non-symmetric $2$-primal ring
O'Meara's infinite matrix algebra
Perfect ring that isn't semiprimary
Puninski's triangular serial 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 Artinian triangular ring
Right-not-left coherent ring
Right-not-left Kasch ring
Right-not-left nonsingular ring
Right-not-left simple injective ring
Semicommutative $R$ such that $R[x]$ is not semicommutative
Simple, Noetherian ring with zero divisors and trivial idempotents
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
Šter's counterexample showing "clean" is not Morita invariant
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Z[X]/(X^2,4X, 8)$
$C([0,1])$, the ring of continuous real-valued functions on the unit interval
$F_2[x,y]/(x,y)^2$
$T_n(F_2)$
Cohn's non-IBN domain
Hochster's connected, nondomain, locally-domain ring
Kasch not semilocal ring
Malcev's nonembeddable domain
Osofsky's $32$ element ring
Perfect non-Artinian ring
Progression free polynomial ring
Ram's Ore extension ring
Reversible non-symmetric ring
Right-not-left Noetherian triangular ring
Shepherdson's domain that is not stably finite
$2$-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
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb R$: the field of real numbers
$\mathbb R[[x]]$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x,y]$ 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
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\prod_{i=1}^\infty F_2$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$^\ast \mathbb R$: the field of hyperreal numbers
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_p(x)$
$k[x,x^{-1};\sigma]$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
$M_n(\mathbb Q)$
$M_n(F_2)$
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
Bergman's example showing that "compressible" is not Morita invariant
catenary, not universally catenary
Clark's uniserial 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
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
Finitely cogenerated, not semilocal ring
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 $\Bbb Z_{(2)}$
Hurwitz quaternions
Interval monoid ring
Kolchin's simple V-domain
Left-not-right Noetherian domain
Left-not-right pseudo-Frobenius ring
Left-not-right uniserial domain
Lipschitz quaternions
Mori but not Krull domain
Nagata ring that not quasi-excellent
Nagata's Noetherian infinite Krull dimension ring
Nakayama's quasi-Frobenius ring that isn't Frobenius
Noetherian domain that is not N-1
Noetherian ring that is not Grothendieck and not Nagata
non-$h$-local domain
Nonlocal endomorphism ring of a uniserial module
Osofsky's Type I ring
Page's left-not-right FPF ring
Pseudo-Frobenius, not quasi-Frobenius ring
Quasi-continuous ring that is not Ikeda-Nakayama
Rational quaternions
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
Legend
• = has the property
• = does not have the property
• = information not in database