Property: (right/left) distributive

Definition: (right distributive) Lattice of right ideals is a distributive lattice

Reference(s):

(No citations retrieved.)

Metaproperties:

This property has the following metaproperties
  • passes to quotient rings
  • stable under products
  • stable under finite products
  • forms an equational class
This property does not have the following metaproperties
Rings
left Name right
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[x]/(x^2-1)$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$C([0,1])$, 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 $0$
$k[[x,y]]/(x^2,xy)$
$k[x,x^{-1};\sigma]$
Algebra of differential operators on the line (1st Weyl algebra)
Basic ring of Nakayama's QF ring
Bergman's example showing that "compressible" is not Morita invariant
Bergman's primitive finite uniform dimension ring
Bergman's right-not-left primitive ring
Bergman's ring with IBN
Bergman's ring without IBN
Camillo and Nielsen's McCoy ring
Cohn's right-not-left free ideal ring
Cozzens simple, left principal, right non-Noetherian domain
Cozzens' simple V-domain
Eventually constant sequences in $\mathbb Z$
Faith-Menal counterexample
Grams' atomic domain which doesn't satisfy ACCP
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
Hochster's connected, nondomain, locally-domain ring
Hurwitz quaternions
Kasch not semilocal ring
Kerr's Goldie ring with non-Goldie matrix 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
Lipschitz quaternions
Local right-not-left Kasch ring
McGovern's commutative Zorn ring that isn't clean
Mori but not Krull domain
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
Nielsen's semicommutative ring that isn't McCoy
Non-Artinian simple ring
Page's left-not-right FPF ring
Progression free polynomial ring
Pseudo-Frobenius, not quasi-Frobenius ring
Puninski's triangular serial ring
Quasi-continuous ring that is not Ikeda-Nakayama
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 coherent ring
Semicommutative $R$ such that $R[x]$ is not semicommutative
Šter's counterexample showing "clean" is not Morita invariant
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
$\mathbb R[x_1, x_2,x_3,\ldots]$
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\varinjlim T_{2^n}(\Bbb Q)$
$\varinjlim M_{2^n}(\mathbb Q)$
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_2[x,y]/(x,y)^2$
$M_n(\mathbb Q)$
$M_n(F_2)$
$RCFM_\omega(\mathbb Q)$
$T_\omega(\mathbb Q)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
$T_n(F_2)$
$T_n(F_q)$
Akizuki's counterexample
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 unit-regular ring
catenary, not universally catenary
Chase's left-not-right semihereditary ring
Cohn's non-IBN domain
Cohn's Schreier domain that isn't GCD
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
Kaplansky's right-not-left hereditary ring
Malcev's nonembeddable domain
Michler & Villamayor's right-not-left V ring
Nagata ring that not quasi-excellent
Nagata's Noetherian infinite Krull dimension ring
Noetherian domain that is not N-1
Non-symmetric $2$-primal ring
O'Meara's infinite matrix algebra
Osofsky's $32$ element ring
Perfect non-Artinian ring
Perfect ring that isn't semiprimary
Ram's Ore extension ring
Reversible non-symmetric ring
Right-not-left Artinian triangular 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 $\mathbb C^n$, $n>1$
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
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: $\mathbb Z_2$
$\mathbb C$: 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
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x]$
$\mathbb R$: the field of real numbers
$\mathbb R[[x]]$
$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
$\mathbb R[x]/(x^2)$
$\mathbb Z$: 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
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\prod_{i=1}^\infty F_2$
$\varinjlim \mathbb Q^{2^n}$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$^\ast \mathbb R$: the field of hyperreal numbers
$F_p(x)$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
10-adic numbers
2-dimensional uniserial domain
2-truncated Witt vectors over $\Bbb F_2((t))$
Algebraic closure of $F_2$
Algebraic integers
Clark's uniserial ring
Countably infinite boolean ring
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
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
Finitely cogenerated, not semilocal ring
Henselization of $\Bbb Z_{(2)}$
Interval monoid ring
Left-not-right uniserial domain
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
Rational quaternions
Ring of holomorphic functions on $\mathbb C$
Square of a torch ring
Trivial extension torch ring
Legend
  • = has the property
  • = does not have the property
  • = information not in database