DaRT - Ring Property 101

Rings

left	Name	right
	$2$-adic integers: $\mathbb Z_2$
	$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
	$\mathbb C$: 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
	$\mathbb Q+FM_\omega(\mathbb Q)$
	$\mathbb Q[[x^2,x^3]]$
	$\mathbb Q[\mathbb Q]$
	$\mathbb Q[x,x^{-1}]$: Laurent polynomials
	$\mathbb Q[x,y,z]/(xz,yz)$
	$\mathbb Q[x,y]/(x^2, xy)$
	$\mathbb Q[x,y]/(x^2-y^3)$
	$\mathbb Q[X,Y]_{(X,Y)}$
	$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
	$\mathbb Q[x]$
	$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
	$\mathbb Q[x_1, x_2,\ldots, x_n]$
	$\mathbb Q\langle a,b\rangle/(a^2)$
	$\mathbb Q\langle x, y\rangle$
	$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
	$\mathbb R$: 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 $S=((2)\cup(3))^c$
	$\mathbb Z_{(2)}$
	$\prod_{i=0}^\infty \mathbb Q$
	$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
	$\prod_{i=1}^\infty \mathbb Z/(2^i)$
	$\prod_{i=1}^\infty F_2$
	$\varinjlim T_{2^n}(\Bbb Q)$
	$\varinjlim \mathbb Q^{2^n}$
	$\varinjlim M_{2^n}(\mathbb Q)$
	$\widehat{\mathbb Z}$: the profinite completion of the integers
	$^\ast \mathbb R$: 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 $0$
	$F_2[\mathcal Q_8]$
	$F_2[S_4]$
	$F_2[x,y]/(x,y)^2$
	$F_p(x)$
	$k[[x,y]]/(x^2,xy)$
	$k[x,x^{-1};\sigma]$
	$k[x;\sigma]/(x^2)$ (Artinian)
	$k[x;\sigma]/(x^2)$ (not right Artinian)
	$M_n(\mathbb Q)$
	$M_n(F_2)$
	$RCFM_\omega(\mathbb Q)$
	$T_\omega(\mathbb Q)$
	$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
	$T_n(F_2)$
	$T_n(F_q)$
	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 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
	catenary, not universally catenary
	Chase's left-not-right semihereditary ring
	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
	Countably infinite boolean ring
	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
	Eventually constant sequences in $\mathbb Z$
	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
	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
	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
	Kaplansky's right-not-left hereditary ring
	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
	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
	Michler & Villamayor's right-not-left V ring
	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-$h$-local domain
	Non-Artinian simple ring
	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
	Page's left-not-right FPF ring
	Perfect non-Artinian ring
	Perfect ring that isn't semiprimary
	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
	Rational quaternions
	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 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
	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
	Square of a torch ring
	Trivial extension torch ring
	Varadarajan's left-not-right coHopfian ring
	Šter's clean ring with non-clean corner rings
	Šter's counterexample showing "clean" is not Morita invariant

Property: (right/left) essential socle

Reference(s):

Metaproperties: