Property: IC ring

Definition: "Internal Cancellation": If $A\oplus B=R$ and $A'\oplus B'=R$ are two decompositions of $R$ into right ideals, and if $A\cong A'$, then also $B\cong B'$. (This condition turns out to be left-right symmetric.)

Reference(s):

  • D. Khurana and T. Lam. Rings with internal cancellation. (2005) @ (entire article)

Metaproperties:

This property has the following metaproperties
  • stable under products
  • stable under finite products
  • passes to power series ring
  • passes to $eRe$ for any idempotent $e$
  • passes to $eRe$ for any full idempotent $e$
This property does not have the following metaproperties
  • passes to quotient rings (Counterexample: $R_{ 119 }$ is a homomorphic image of $R_{ 165 }$)
  • passes to subrings (Counterexample: $R_{ 82 }$ is a subring of $R_{ 81 }$)
  • passes to matrix rings (counterexample needed)
  • passes to polynomial rings (counterexample needed)
  • Morita invariant (counterexample needed)
Rings
Name
$2$-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
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[[X]]$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: 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
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 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
Domanov's prime, nonprimitive, von Neumann regular ring
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
McCoy ring that is not Abelian
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 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-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
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
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
Legend
  • = has the property
  • = does not have the property
  • = information not in database