DaRT - Ring Property 101

Rings

left	Name	right
	$\mathbb Q+FM_\omega(\mathbb Q)$
	$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
	$\mathbb Q\langle a,b\rangle/(a^2)$
	$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
	$\mathbb Z\langle x,y\rangle/(y^2, yx)$
	$\prod_{i=0}^\infty \mathbb Q$
	$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
	$RCFM_\omega(\mathbb Q)$
	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
	Chase's left-not-right semihereditary ring
	Eventually constant sequences in $\mathbb Z$
	Facchini's torch ring
	Goodearl's simple self-injective operator algebra
	Goodearl's simple self-injective von Neumann regular ring
	Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
	Hochster's connected, nondomain, locally-domain ring
	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
	McGovern's commutative Zorn ring that isn't clean
	Nielsen's right UGP, not left UGP ring
	Nielsen's semicommutative ring that isn't McCoy
	Non-symmetric $2$-primal ring
	Nonlocal endomorphism ring of a uniserial module
	O'Meara's infinite matrix algebra
	Page's left-not-right FPF ring
	Progression free polynomial ring
	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
	Semicommutative $R$ such that $R[x]$ is not semicommutative
	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
	$2$-adic integers: $\mathbb Z_2$
	$\mathbb Q[[x^2,x^3]]$
	$\mathbb Q[\mathbb Q]$
	$\mathbb Q[x,x^{-1}]$: Laurent polynomials
	$\mathbb Q[x,y,z]/(xz,yz)$
	$\mathbb Q[x,y]/(x^2, xy)$
	$\mathbb Q[x,y]/(x^2-y^3)$
	$\mathbb Q[X,Y]_{(X,Y)}$
	$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
	$\mathbb Q[x]$
	$\mathbb Q[x_1, x_2,\ldots, x_n]$
	$\mathbb Q\langle x, y\rangle$
	$\mathbb R[[x]]$
	$\mathbb R[x,y]$ 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
	$\mathbb Z[x]$
	$\mathbb Z[x]/(x^2-1)$
	$\mathbb Z[x_0, x_1,x_2,\ldots]$
	$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
	$\mathbb Z_S$, where $S=((2)\cup(3))^c$
	$\mathbb Z_{(2)}$
	$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
	$\varinjlim T_{2^n}(\Bbb Q)$
	$\varinjlim \mathbb Q^{2^n}$
	$\varinjlim M_{2^n}(\mathbb Q)$
	$\widehat{\mathbb Z}$: the profinite completion of the integers
	$C([0,1])$, the ring of continuous real-valued functions on the unit interval
	$k[[x,y]]/(x^2,xy)$
	$k[x,x^{-1};\sigma]$
	10-adic numbers
	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 primitive finite uniform dimension ring
	Bergman's right-not-left primitive 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
	Grams' atomic domain which doesn't satisfy ACCP
	Henselization of $\Bbb Z_{(2)}$
	Hurwitz quaternions
	Kolchin's simple V-domain
	Left-not-right Noetherian domain
	Left-not-right uniserial domain
	Lipschitz quaternions
	Local right-not-left Kasch ring
	Malcev's nonembeddable domain
	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
	Noetherian domain that is not N-1
	Noetherian ring that is not Grothendieck and not Nagata
	non-$h$-local domain
	Non-Artinian simple ring
	Osofsky's Type I ring
	Ram's Ore extension 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
	Simple, non-Artinian, von Neumann regular ring
	$\mathbb A_\mathbb Q$: 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
	$\mathbb R$: 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, $k>1$
	$\mathbb Z[X]/(X^2,4X, 8)$
	$\mathbb Z[X]/(X^2,8)$
	$\prod_{i=1}^\infty \mathbb Z/(2^i)$
	$\prod_{i=1}^\infty F_2$
	$^\ast \mathbb R$: the field of hyperreal numbers
	$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
	$F_2[\mathcal Q_8]$
	$F_2[S_4]$
	$F_2[x,y]/(x,y)^2$
	$F_p(x)$
	$k[x;\sigma]/(x^2)$ (Artinian)
	$k[x;\sigma]/(x^2)$ (not right Artinian)
	$M_n(\mathbb Q)$
	$M_n(F_2)$
	$T_\omega(\mathbb Q)$
	$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
	$T_n(F_2)$
	$T_n(F_q)$
	2-truncated Witt vectors over $\Bbb F_2((t))$
	Algebraic closure of $F_2$
	Basic ring of Nakayama's QF ring
	Bass's right-not-left perfect ring
	Berberian's incompressible Baer ring
	Bergman's exchange ring that isn't clean
	Clark's uniserial ring
	Countably infinite boolean ring
	Division algebra with no anti-automorphism
	Division ring with an antihomomorphism but no involution
	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
	Interval monoid ring
	Kasch not semilocal ring
	Left-not-right pseudo-Frobenius ring
	Michler & Villamayor's right-not-left V ring
	Nakayama's quasi-Frobenius ring that isn't Frobenius
	Osofsky's $32$ element ring
	Perfect non-Artinian ring
	Perfect ring that isn't semiprimary
	Pseudo-Frobenius, not quasi-Frobenius ring
	Quasi-continuous ring that is not Ikeda-Nakayama
	Rational quaternions
	Reversible non-symmetric ring
	Right-not-left Artinian triangular ring
	Right-not-left coherent ring
	Right-not-left Kasch ring
	Right-not-left simple injective ring

Property: (right/left) essential socle

Reference(s):

Metaproperties: