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Property: (right/left) semihereditary
Definition: (right semihereditary) All finitely generated right ideals are projective
Reference(s):
T.-Y. Lam. Lectures on modules and rings. (2012) @ Sections 2E, 2G, 4H, 7E
W. Dicks and A. Schofield. On semihereditary rings. (1988) @ (whole paper)
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
passes to quotient rings (Counterexample:
$R_{ 49 }$
is a homomorphic image of
$R_{ 27 }$
)
passes to subrings (Counterexample:
$R_{ 6 }$
is a subring of
$R_{ 101 }$
)
Rings
left
Name
right
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q\langle x, y\rangle$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb Z[x_0, x_1,x_2,\ldots]$
$\mathbb Z\langle x_0, x_1,x_2,\ldots\rangle$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\varinjlim T_{2^n}(\Bbb Q)$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$k[x,x^{-1};\sigma]$
$RCFM_\omega(\mathbb Q)$
$T_\omega(\mathbb Q)$
10-adic numbers
Berberian's incompressible Baer ring
Bergman's example showing that "compressible" is not Morita invariant
Bergman's right-not-left primitive ring
Bergman's ring with IBN
Bergman's ring without IBN
Camillo and Nielsen's McCoy ring
Cohn's non-IBN domain
Eventually constant sequences in $\mathbb Z$
Faith-Menal counterexample
Grams' atomic domain which doesn't satisfy ACCP
Kerr's Goldie ring with non-Goldie matrix ring
Leavitt path algebra of an infinite bouquet of circles
Lipschitz quaternions
Malcev's nonembeddable domain
McGovern's commutative Zorn ring that isn't clean
Mori but not Krull domain
Nagata's normal ring that is not analytically normal
Nielsen's semicommutative ring that isn't McCoy
Non-Artinian simple ring
Non-symmetric $2$-primal ring
O'Meara's infinite matrix algebra
Page's left-not-right FPF ring
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
Šter's counterexample showing "clean" is not Morita invariant
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$
$\mathbb R[x]/(x^2)$
$\mathbb R[x_1, x_2,x_3,\ldots]$
$\mathbb Z/(n)$, $n$ divisible by two primes and a square
$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
$\mathbb Z[x]$
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z[X]/(X^2,8)$
$\mathbb Z[x]/(x^2-1)$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$C([0,1])$, 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
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
Chase's left-not-right semihereditary ring
Clark's uniserial ring
Cohn's Schreier domain that isn't GCD
Facchini's torch ring
Finitely cogenerated, not semilocal ring
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
Hochster's connected, nondomain, locally-domain ring
Interval monoid ring
Kasch not semilocal ring
Left-not-right pseudo-Frobenius ring
Local right-not-left Kasch ring
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
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 nonsingular ring
Right-not-left simple injective ring
ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$
Samuel's UFD having a non-UFD power series ring
Simple, Noetherian ring with zero divisors and trivial idempotents
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
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: 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
$\mathbb Z+x\mathbb Q[x]$
$\mathbb Z/(2)$
$\mathbb Z/(n)$, $n$ 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 $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty F_2$
$\varinjlim \mathbb Q^{2^n}$
$\varinjlim M_{2^n}(\mathbb Q)$
$^\ast \mathbb R$: 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 $\mathbb Q$
$T_n(F_2)$
$T_n(F_q)$
2-dimensional uniserial domain
Algebra of differential operators on the line (1st Weyl algebra)
Algebraic closure of $F_2$
Algebraic integers
Bergman's exchange ring that isn't clean
Bergman's non-unit-regular subring
Bergman's unit-regular ring
Cohn's right-not-left free ideal ring
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
field of $2$-adic numbers
Field of algebraic numbers
Field of constructible numbers
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
Henselization of $\Bbb Z_{(2)}$
Hurwitz quaternions
Kaplansky's right-not-left hereditary ring
Kolchin's simple V-domain
Left-not-right Noetherian domain
Left-not-right uniserial domain
Michler & Villamayor's right-not-left V ring
Noetherian ring that is not Grothendieck and not Nagata
non-$h$-local domain
Osofsky's Type I ring
Rational quaternions
Right-not-left Noetherian triangular ring
Ring of holomorphic functions on $\mathbb C$
Simple, non-Artinian, von Neumann regular ring
Small's right hereditary, not-left semihereditary ring
Šter's clean ring with non-clean corner rings
Legend
= has the property
= does not have the property
= information not in database