Property: (right/left) continuous

Definition: (right continuous) $R$ is right CS, and any right ideal isomorphic to a direct summand of $R$ is itself a summand.


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This property does not have the following metaproperties
left Name right
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\varinjlim T_{2^n}(\Bbb Q)$
$\varinjlim \mathbb Q^{2^n}$
$\varinjlim M_{2^n}(\mathbb Q)$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$T_\omega(\mathbb Q)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
10-adic numbers
Bass's right-not-left perfect ring
Berberian's incompressible Baer ring
Bergman's non-unit-regular subring
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
Countably infinite boolean ring
Eventually constant sequences in $\mathbb Z$
Faith-Menal counterexample
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
Kaplansky's right-not-left hereditary ring
Kerr's Goldie ring with non-Goldie matrix ring
Left-not-right pseudo-Frobenius ring
Michler & Villamayor's right-not-left V ring
Nagata's normal ring that is not analytically normal
Nielsen's semicommutative ring that isn't McCoy
Non-Artinian simple ring
Page's left-not-right FPF ring
Perfect ring that isn't semiprimary
Puninski's triangular serial ring
Right-not-left Artinian triangular ring
Right-not-left Kasch ring
Right-not-left simple injective ring
Semicommutative $R$ such that $R[x]$ is not semicommutative
Simple, non-Artinian, von Neumann regular ring
Small's right hereditary, not-left semihereditary ring
Varadarajan's left-not-right coHopfian ring
Šter's clean ring with non-clean corner rings
$2$-adic integers: $\mathbb Z_2$
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: 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
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$C([0,1])$, the ring of continuous real-valued functions on the unit interval
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
$RCFM_\omega(\mathbb Q)$
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 primitive finite uniform dimension ring
catenary, not universally catenary
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
DVR that is not N-2
Facchini's torch ring
Finitely cogenerated, not semilocal ring
Grams' atomic domain which doesn't satisfy ACCP
Henselization of $\Bbb Z_{(2)}$
Hochster's connected, nondomain, locally-domain ring
Hurwitz quaternions
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 uniserial domain
Lipschitz quaternions
Local right-not-left Kasch ring
Malcev's nonembeddable domain
McGovern's commutative Zorn ring that isn't clean
Mori but not Krull domain
Nagata ring that not quasi-excellent
Nagata's Noetherian infinite Krull dimension ring
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
O'Meara's infinite matrix algebra
Osofsky's $32$ element ring
Osofsky's Type I ring
Perfect non-Artinian ring
Progression free polynomial ring
Ram's Ore extension ring
reduced $I_0$ ring that is not exchange
reduced exchange ring which is not semiregular
Reversible non-symmetric ring
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, Noetherian ring with zero divisors and trivial idempotents
Square of a torch ring
Trivial extension torch ring
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, $k>1$
$\mathbb Z[X]/(X^2,8)$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\prod_{i=1}^\infty F_2$
$^\ast \mathbb R$: the field of hyperreal numbers
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_2[\mathcal Q_8]$
$M_n(\mathbb Q)$
2-truncated Witt vectors over $\Bbb F_2((t))$
Algebraic closure of $F_2$
Basic ring of Nakayama's QF ring
Clark's uniserial 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
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
Interval monoid ring
Nakayama's quasi-Frobenius ring that isn't Frobenius
Pseudo-Frobenius, not quasi-Frobenius ring
Quasi-continuous ring that is not Ikeda-Nakayama
Rational quaternions
Right-not-left coherent ring
  • = has the property
  • = does not have the property
  • = information not in database