DaRT - Ring Property 74

Rings

left	Name	right
	$\mathbb Q\langle a,b\rangle/(a^2)$
	$\mathbb Q\langle x, y\rangle$
	$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
	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 ring with IBN
	Bergman's ring without IBN
	Bergman's unit-regular ring
	Camillo and Nielsen's McCoy ring
	Cohn's non-IBN domain
	Cohn's right-not-left free ideal ring
	Faith-Menal counterexample
	Hurwitz quaternions
	Kaplansky's right-not-left hereditary ring
	Left-not-right Noetherian domain
	Lipschitz quaternions
	Malcev's nonembeddable domain
	Michler & Villamayor's right-not-left V ring
	Right-not-left ACC on annihilators triangular ring
	Shepherdson's domain that is not stably finite
	Small's right hereditary, not-left semihereditary ring
	Šter's clean ring with non-clean corner rings
	Šter's counterexample showing "clean" is not Morita invariant
	$2$-adic integers: $\mathbb Z_2$
	$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
	$\mathbb Q[[X]]$
	$\mathbb Q[[x^2,x^3]]$
	$\mathbb Q[\mathbb Q]$
	$\mathbb Q[x,x^{-1}]$: Laurent polynomials
	$\mathbb Q[x,y,z]/(xz,yz)$
	$\mathbb Q[x,y]/(x^2, xy)$
	$\mathbb Q[x,y]/(x^2-y^3)$
	$\mathbb Q[X,Y]_{(X,Y)}$
	$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
	$\mathbb Q[x]$
	$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
	$\mathbb Q[x_1, x_2,\ldots, x_n]$
	$\mathbb R[[x]]$
	$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
	$\mathbb R[x,y]$ 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
	$\mathbb Z+x\mathbb Q[x]$
	$\mathbb Z/(n)$, $n$ divisible by two primes and a square
	$\mathbb Z/(n)$, $n$ squarefree and not 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
	$\mathbb Z[x]$
	$\mathbb Z[X]/(X^2,4X, 8)$
	$\mathbb Z[X]/(X^2,8)$
	$\mathbb Z[x]/(x^2-1)$
	$\mathbb Z[x_0, x_1,x_2,\ldots]$
	$\mathbb Z\langle x,y\rangle/(y^2, yx)$
	$\mathbb Z_S$, where $S=((2)\cup(3))^c$
	$\mathbb Z_{(2)}$
	$\prod_{i=0}^\infty \mathbb Q$
	$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
	$\prod_{i=1}^\infty \mathbb Z/(2^i)$
	$\prod_{i=1}^\infty F_2$
	$\varinjlim T_{2^n}(\Bbb Q)$
	$\varinjlim \mathbb Q^{2^n}$
	$\widehat{\mathbb Z}$: the profinite completion of the integers
	$C([0,1])$, 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 $0$
	$F_2[\mathcal Q_8]$
	$F_2[S_4]$
	$F_2[x,y]/(x,y)^2$
	$k[[x,y]]/(x^2,xy)$
	$k[x;\sigma]/(x^2)$ (Artinian)
	$k[x;\sigma]/(x^2)$ (not right Artinian)
	$T_\omega(\mathbb Q)$
	$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
	$T_n(F_2)$
	$T_n(F_q)$
	10-adic numbers
	2-dimensional uniserial domain
	2-truncated Witt vectors over $\Bbb F_2((t))$
	Akizuki's counterexample
	Algebraic integers
	Base ring for $R_{187}$
	Base ring for $R_{191}$
	Basic ring of Nakayama's QF ring
	Bass's right-not-left perfect ring
	Berberian's incompressible Baer ring
	Bergman's right-not-left primitive ring
	catenary, not universally catenary
	Chase's left-not-right semihereditary ring
	Clark's uniserial ring
	Cohn's Schreier domain that isn't GCD
	Countably infinite boolean ring
	Custom Krull dimension valuation ring
	Domanov's prime, nonprimitive, von Neumann regular ring
	DVR that is not N-2
	Eventually constant sequences in $\mathbb Z$
	Facchini's torch ring
	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
	Interval monoid ring
	Kasch not semilocal ring
	Kerr's Goldie ring with non-Goldie matrix ring
	Left-not-right pseudo-Frobenius ring
	Left-not-right uniserial domain
	Local right-not-left Kasch ring
	McCoy ring that is not Abelian
	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
	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
	Noetherian domain that is not N-1
	Noetherian ring that is not Grothendieck and not Nagata
	Non lift/rad matrix ring over a lift/rad base ring
	Non π-regular matrix ring over a π-regular ring
	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
	Page's left-not-right FPF ring
	Perfect non-Artinian ring
	Perfect ring that isn't semiprimary
	prime, von Neumann regular, nonprimitive Leavitt path algebra
	Progression free polynomial ring
	Pseudo-Frobenius, not quasi-Frobenius ring
	Puninski's triangular serial ring
	Quasi-continuous ring that is not Ikeda-Nakayama
	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 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
	Semicommutative $R$ such that $R[x]$ is not semicommutative
	Square of a torch ring
	Trivial extension torch ring
	Varadarajan's left-not-right coHopfian ring
	$\mathbb C$: 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
	$\mathbb Q+FM_\omega(\mathbb Q)$
	$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
	$\mathbb R$: the field of real numbers
	$\mathbb Z/(2)$
	$\mathbb Z/(p)$, $p$ an odd prime
	$\varinjlim M_{2^n}(\mathbb Q)$
	$^\ast \mathbb R$: the field of hyperreal numbers
	$F_p(x)$
	$k[x,x^{-1};\sigma]$
	$M_n(\mathbb Q)$
	$M_n(F_2)$
	$RCFM_\omega(\mathbb Q)$
	Algebra of differential operators on the line (1st Weyl algebra)
	Algebraic closure of $F_2$
	Bergman's primitive finite uniform dimension ring
	Cozzens simple, left principal, right non-Noetherian domain
	Cozzens' simple V-domain
	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
	Kolchin's simple V-domain
	Leavitt path algebra of an infinite bouquet of circles
	Non-Artinian simple ring
	O'Meara's infinite matrix algebra
	Rational quaternions
	Simple, Noetherian ring with zero divisors and trivial idempotents
	Simple, non-Artinian, von Neumann regular ring

Property: (right/left) primitive

Reference(s):

Metaproperties: