Property: (right/left) principally injective

Definition: (right principally injective) homomorphisms from principal right ideals of the ring into the ring extend to endomorphisms of the ring


  • E. A. Rutter and Jr. Rings with the principal extension property. (1975) @ .
  • W. K. Nicholson and M. F. Yousif. Principally injective rings. (1995) @ .


This property has the following metaproperties
  • stable under products
  • stable under finite products
This property does not have the following metaproperties
left Name right
$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra
$\mathbb Z[X]/(X^2,4X, 8)$
$\mathbb Z\langle x,y\rangle/(y^2, yx)$
$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$
$\varinjlim T_{2^n}(\Bbb Q)$
$\widehat{\mathbb Z}$: the profinite completion of the integers
$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
$RCFM_\omega(\mathbb Q)$
$T_\omega(\mathbb Q)$
$T_n(\mathbb Q)$: the upper triangular matrix ring over $\mathbb Q$
Bass's right-not-left perfect ring
Berberian's incompressible Baer ring
Bergman's primitive finite uniform dimension ring
Bergman's right-not-left primitive ring
Bergman's ring with IBN
Bergman's ring without IBN
Camillo and Nielsen's McCoy ring
Chase's left-not-right semihereditary ring
Eventually constant sequences in $\mathbb Z$
Facchini's torch ring
Grassmann algebra $\bigwedge (V)$, $\dim(V)=\aleph_0$
Kasch not semilocal ring
Leavitt path algebra of an infinite bouquet of circles
McGovern's commutative Zorn ring that isn't clean
Nielsen's semicommutative ring that isn't McCoy
Non-Artinian simple ring
Nonlocal endomorphism ring of a uniserial module
O'Meara's infinite matrix algebra
Page's left-not-right FPF ring
Perfect non-Artinian ring
Perfect ring that isn't semiprimary
Puninski's triangular serial ring
reduced $I_0$ ring that is not exchange
reduced exchange ring which is not semiregular
Reversible non-symmetric ring
Right-not-left Kasch ring
Right-not-left simple injective 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
Šter's counterexample showing "clean" is not Morita invariant
$2$-adic integers: $\mathbb Z_2$
$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
$\mathbb Q[[x^2,x^3]]$
$\mathbb Q[\mathbb Q]$
$\mathbb Q[x,x^{-1}]$: Laurent polynomials
$\mathbb Q[x,y,z]/(xz,yz)$
$\mathbb Q[x,y]/(x^2, xy)$
$\mathbb Q[x,y]/(x^2-y^3)$
$\mathbb Q[X,Y]_{(X,Y)}$
$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
$\mathbb Q[x]$
$\mathbb Q[x_1, x_2,\ldots, x_n]$
$\mathbb Q\langle a,b\rangle/(a^2)$
$\mathbb Q\langle x, y\rangle$
$\mathbb R[[x]]$
$\mathbb R[x,y]$ 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 $S=((2)\cup(3))^c$
$\mathbb Z_{(2)}$
$C([0,1])$, the ring of continuous real-valued functions on the unit interval
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
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
Faith-Menal counterexample
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
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
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
Nielsen's right UGP, not left UGP ring
Noetherian domain that is not N-1
Noetherian ring that is not Grothendieck and not Nagata
non-$h$-local domain
Non-symmetric $2$-primal ring
Osofsky's $32$ element ring
Osofsky's Type I ring
Progression free polynomial ring
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 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
Varadarajan's left-not-right coHopfian ring
$\mathbb C$: 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
$\mathbb Q+FM_\omega(\mathbb Q)$
$\mathbb R$: 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, $k>1$
$\mathbb Z[X]/(X^2,8)$
$\prod_{i=0}^\infty \mathbb Q$
$\prod_{i=1}^\infty \mathbb Z/(2^i)$
$\prod_{i=1}^\infty F_2$
$\varinjlim \mathbb Q^{2^n}$
$\varinjlim M_{2^n}(\mathbb Q)$
$^\ast \mathbb R$: the field of hyperreal numbers
$C\ell_{2,1}(\mathbb R)$: the geometric algebra of Minkowski 3-space
$F_2[\mathcal Q_8]$
$k[x;\sigma]/(x^2)$ (Artinian)
$k[x;\sigma]/(x^2)$ (not right Artinian)
$M_n(\mathbb Q)$
2-truncated Witt vectors over $\Bbb F_2((t))$
Algebraic closure of $F_2$
Basic ring of Nakayama's QF ring
Bergman's exchange ring that isn't clean
Bergman's non-unit-regular subring
Bergman's unit-regular ring
Clark's uniserial ring
Countably infinite boolean ring
Division algebra with no anti-automorphism
Division ring with an antihomomorphism but no involution
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
Interval monoid ring
Kaplansky's right-not-left hereditary ring
Left-not-right pseudo-Frobenius ring
Michler & Villamayor's right-not-left V ring
Nakayama's quasi-Frobenius ring that isn't Frobenius
Pseudo-Frobenius, not quasi-Frobenius ring
Quasi-continuous ring that is not Ikeda-Nakayama
Rational quaternions
Simple, non-Artinian, von Neumann regular ring
Šter's clean ring with non-clean corner rings
  • = has the property
  • = does not have the property
  • = information not in database