Property: (right/left) hereditary

Definition: (right hereditary) All right ideals are projective

Reference(s):

  • T.-Y. Lam. Lectures on modules and rings. (2012) @ Sections 2E, 2G, 4H, 7E

Metaproperties:

This property has the following metaproperties
  • Morita invariant
  • passes to matrix rings
  • passes to $eRe$ for any full idempotent $e$
This property does not have the following metaproperties
Rings
left Name right
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\varinjlim T_{2^n}(\Bbb Q)$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$k[x,x^{-1};\sigma]$
$RCFM_\omega(\mathbb Q)$
$T_\omega(\mathbb Q)$
10-adic numbers
2-dimensional uniserial domain
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 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
Chase's left-not-right semihereditary ring
Cohn's non-IBN domain
Cohn's right-not-left free ideal ring
Eventually constant sequences in $\mathbb Z$
Faith-Menal counterexample
Full linear ring of a countable dimensional right vector space
Leavitt path algebra of an infinite bouquet of circles
Lipschitz quaternions
Malcev's nonembeddable domain
McGovern's commutative Zorn ring that isn't clean
Michler & Villamayor's right-not-left V ring
Nagata's normal ring that is not analytically normal
Nielsen's semicommutative ring that isn't McCoy
non-$h$-local domain
Non-Artinian simple ring
Non-symmetric $2$-primal ring
O'Meara's infinite matrix algebra
reduced $I_0$ ring that is not exchange
reduced exchange ring which is not semiregular
Right-not-left ACC on annihilators triangular ring
Right-not-left Artinian triangular ring
Semicommutative $R$ such that $R[x]$ is not semicommutative
Shepherdson's domain that is not stably finite
Simple, non-Artinian, von Neumann regular ring
Šter's clean ring with non-clean corner rings
Šter's counterexample showing "clean" is not Morita invariant
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
$\mathbb R[x]/(x^2)$
$\mathbb R[x_1, x_2,x_3,\ldots]$
$\mathbb Z+x\mathbb Q[x]$
$\mathbb Z/(n)$, $n$ divisible by two primes and a square
$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\prod_{i=1}^\infty F_2$
$C([0,1])$, 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 $0$
$F_2[\mathcal Q_8]$
$F_2[S_4]$
$F_2[x,y]/(x,y)^2$
$k[[x,y]]/(x^2,xy)$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
2-truncated Witt vectors over $\Bbb F_2((t))$
Akizuki's counterexample
Algebraic integers
Basic ring of Nakayama's QF ring
Bass's right-not-left perfect ring
Bergman's primitive finite uniform dimension ring
catenary, not universally catenary
Clark's uniserial ring
Cohn's Schreier domain that isn't GCD
Custom Krull dimension valuation ring
Facchini's torch ring
Finitely cogenerated, not semilocal ring
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 $\bigwedge (V)$, $\dim(V)=\aleph_0$
Hochster's connected, nondomain, locally-domain ring
Interval monoid ring
Kaplansky's right-not-left hereditary ring
Kasch not semilocal ring
Kerr's Goldie ring with non-Goldie matrix ring
Left-not-right pseudo-Frobenius ring
Local right-not-left Kasch ring
Mori but not Krull domain
Nagata ring that not quasi-excellent
Nagata's Noetherian infinite Krull dimension ring
Nakayama's quasi-Frobenius ring that isn't Frobenius
Nielsen's right UGP, not left UGP ring
Noetherian domain that is not N-1
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
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
Reversible non-symmetric 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
Simple, Noetherian ring with zero divisors and trivial idempotents
Small's right hereditary, not-left semihereditary ring
Square of a torch ring
Trivial extension torch ring
Varadarajan's left-not-right coHopfian ring
$2$-adic integers: $\mathbb Z_2$
$\mathbb C$: 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
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x]$
$\mathbb R$: the field of real numbers
$\mathbb R[[x]]$
$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions
$\mathbb Z$: the ring of integers
$\mathbb Z/(2)$
$\mathbb Z/(n)$, $n$ squarefree and not prime
$\mathbb Z/(p)$, $p$ an odd prime
$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
$\mathbb Z[\sqrt{-5}]$
$\mathbb Z[i]$: the Gaussian integers
$\mathbb Z_S$, where $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$\varinjlim \mathbb Q^{2^n}$
$\varinjlim M_{2^n}(\mathbb Q)$
$^\ast \mathbb R$: the field of hyperreal numbers
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_p(x)$
$M_n(\mathbb Q)$
$M_n(F_2)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
$T_n(F_2)$
$T_n(F_q)$
Algebra of differential operators on the line (1st Weyl algebra)
Algebraic closure of $F_2$
Countably infinite boolean 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
DVR that is not N-2
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
Henselization of $\Bbb Z_{(2)}$
Hurwitz quaternions
Kolchin's simple V-domain
Left-not-right Noetherian domain
Left-not-right uniserial domain
Noetherian ring that is not Grothendieck and not Nagata
Rational quaternions
Legend
  • = has the property
  • = does not have the property
  • = information not in database