Property: (right/left) McCoy

Definition: (right McCoy) A ring $R$ is called right McCoy if when $f,g\in R[x]$ satisfy $fg=0$, then there exists a nonzero $r\in R$ such that $fr=0$.

Reference(s):

  • M. B. Rege, S. Chhawchharia, and others. Armendariz rings. (1997) @ .
  • V. Camillo and P. P. Nielsen. McCoy rings and zero-divisors. (2008) @ (page needed)

Metaproperties:

This property has the following metaproperties
  • stable under products
  • stable under finite products
  • passes to polynomial rings
This property does not have the following metaproperties
Rings
left Name right
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\varinjlim T_{2^n}(\Bbb Q)$
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_2[S_4]$
$T_\omega(\mathbb Q)$
$T_n(F_q)$
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 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
Chase's left-not-right semihereditary ring
Domanov's prime, nonprimitive, von Neumann regular ring
Faith-Menal counterexample
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
Kaplansky's right-not-left hereditary ring
Leavitt path algebra of an infinite bouquet of circles
Left-not-right pseudo-Frobenius ring
Local right-not-left Kasch ring
Michler & Villamayor's right-not-left V ring
Nakayama's quasi-Frobenius ring that isn't Frobenius
Nielsen's right UGP, not left UGP ring
Non lift/rad matrix ring over a lift/rad base ring
Non π-regular matrix ring over a π-regular ring
Non-Artinian simple ring
Non-symmetric $2$-primal ring
Osofsky's $32$ element ring
Page's left-not-right FPF ring
prime, von Neumann regular, nonprimitive Leavitt path algebra
Ram's Ore extension 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
Semicommutative $R$ such that $R[x]$ is not semicommutative
Simple, Noetherian ring with zero divisors and trivial idempotents
Small's right hereditary, not-left semihereditary ring
Varadarajan's left-not-right coHopfian ring
Šter's counterexample showing "clean" is not Morita invariant
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\varinjlim M_{2^n}(\mathbb Q)$
$M_n(\mathbb Q)$
$M_n(F_2)$
$RCFM_\omega(\mathbb Q)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
$T_n(F_2)$
Bergman's exchange ring that isn't clean
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
O'Meara's infinite matrix algebra
Simple, non-Artinian, von Neumann regular 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 $\mathbb Q$
$\mathbb C$: 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
$\mathbb Q[[X]]$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x]$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb R$: 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
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\prod_{i=1}^\infty F_2$
$\varinjlim \mathbb Q^{2^n}$
$\widehat{\mathbb Z}$: 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
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$F_2[\mathcal Q_8]$
$F_2[x,y]/(x,y)^2$
$F_p(x)$
$k[[x,y]]/(x^2,xy)$
$k[x,x^{-1};\sigma]$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
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
Base ring for $R_{187}$
Bergman's example showing that "compressible" is not Morita invariant
Camillo and Nielsen's McCoy ring
catenary, not universally catenary
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
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
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
Interval monoid ring
Kasch not semilocal ring
Kerr's Goldie ring with non-Goldie matrix ring
Kolchin's simple V-domain
Left-not-right Noetherian domain
Left-not-right uniserial domain
Lipschitz quaternions
Malcev's nonembeddable domain
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
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
Nonlocal endomorphism ring of a uniserial module
Osofsky's Type I 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
Rational quaternions
reduced $I_0$ ring that is not exchange
reduced exchange ring which is not semiregular
Reversible non-symmetric 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
Square of a torch ring
Trivial extension torch ring
Legend
  • = has the property
  • = does not have the property
  • = information not in database