Property: complete local

Definition: $R$ is a commutative local ring in which the canonical map into the completion at the maximal ideal is an isomorphism.

Reference(s):

(No citations retrieved.)

Metaproperties:

This property does not have the following metaproperties

passes to subrings (Counterexample: $R_{ 6 }$ is a subring of $R_{ 101 }$)
passes to polynomial rings (Counterexample: $R_{ 7 }$ is the polynomial ring of $R_{ 2 }$)
passes to matrix rings (Counterexample: $R_{ 12 }$ is a matrix ring of $R_{ 2 }$)
Morita invariant (Counterexample: $R_{ 12 }$ is Morita equivalent to $R_{ 2 }$)

	Name
	$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$
	$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$
	$k[[x,y]]/(x^2,xy)$
	ACCP ring with non-ACCP polynomial ring
	Atomic domain with nonatomic polynomial ring
	Clark's uniserial ring
	Cohn's Schreier domain that isn't GCD
	Custom Krull dimension valuation ring
	Grams' atomic domain which doesn't satisfy ACCP
	Mori but not Krull domain
	Non-ACCP polynomial ring
	Non-ACCP power-series
	Non-atomic polynomial ring
	Pseudo-Frobenius, not quasi-Frobenius ring
	Samuel's UFD having a non-UFD power series ring
	$\mathbb A_\mathbb Q$: the ring of adeles of $\mathbb Q$
	$\mathbb Q[\mathbb Q]$
	$\mathbb Q[x,x^{-1}]$: Laurent polynomials
	$\mathbb Q[x,y,z]/(xz,yz)$
	$\mathbb Q[x,y]$
	$\mathbb Q[x,y]/(x^2, xy)$
	$\mathbb Q[x,y]/(x^2-y^3)$
	$\mathbb Q[X,Y]_{(X,Y)}$
	$\mathbb Q[x]$
	$\mathbb Q[x^{1/2},x^{1/4},x^{1/8},...]/(x)$
	$\mathbb Q[x_1, x_2,\ldots, x_n]$
	$\mathbb R[X,Y,Z]/(X^2+Y^2+Z^2-1)$
	$\mathbb R[x,y]$ 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
	$\mathbb Z+x\mathbb Q[x]$
	$\mathbb Z/(n)$, $n$ divisible by two primes and a square
	$\mathbb Z/(n)$, $n$ squarefree and not prime
	$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$
	$\mathbb Z[\sqrt{-5}]$
	$\mathbb Z[i]$: the Gaussian integers
	$\mathbb Z[x]$
	$\mathbb Z[x]/(x^2-1)$
	$\mathbb Z[x_0, x_1,x_2,\ldots]$
	$\mathbb Z_S$, 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
	$C([0,1])$, the ring of continuous real-valued functions on the unit interval
	10-adic numbers
	ACCP ring with non-ACCP power-series
	Akizuki's counterexample
	Algebraic integers
	Base ring for $R_{187}$
	catenary, not universally catenary
	Countably infinite boolean ring
	DVR that is not N-2
	Eventually constant sequences in $\mathbb Q$
	Eventually constant sequences in $\mathbb Z$
	Facchini's torch ring
	Finitely cogenerated, not semilocal ring
	Henselization of $\Bbb Z_{(2)}$
	Hochster's connected, nondomain, locally-domain ring
	Interval monoid ring
	Kasch not semilocal ring
	Kerr's Goldie ring with non-Goldie matrix ring
	McGovern's commutative Zorn ring that isn't clean
	Nagata ring that not quasi-excellent
	Nagata's Noetherian infinite Krull dimension ring
	Nagata's normal ring that is not analytically normal
	Noetherian domain that is not N-1
	Noetherian ring that is not Grothendieck and not Nagata
	non-$h$-local domain
	Noncoherent product of coherent rings
	Osofsky's Type I ring
	Perfect ring that isn't semiprimary
	Principal ideal domain that is not Nagata
	Progression free polynomial ring
	reduced $I_0$ ring that is not exchange
	reduced exchange ring which is not semiregular
	ring of germs of holomorphic functions on $\mathbb C^n$, $n>1$
	Ring of holomorphic functions on $\mathbb C$
	Ring of integer valued polynomials over the rationals
	Square of a torch ring
	Trivial extension torch ring
	$2$-adic integers: $\mathbb Z_2$
	$\mathbb C$: the field of complex numbers
	$\mathbb Q$: the field of rational numbers
	$\mathbb Q(x)$: rational functions over the rational numbers
	$\mathbb Q[[X]]$
	$\mathbb Q[[x^2,x^3]]$
	$\mathbb R$: the field of real numbers
	$\mathbb R[[x]]$
	$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$
	$\mathbb R[x]/(x^2)$
	$\mathbb Z/(2)$
	$\mathbb Z/(p)$, $p$ an odd prime
	$\mathbb Z/(p^k)$, $p$ a prime, $k>1$
	$\mathbb Z[X]/(X^2,4X, 8)$
	$\mathbb Z[X]/(X^2,8)$
	$^\ast \mathbb R$: the field of hyperreal numbers
	$F_2[x,y]/(x,y)^2$
	$F_p(x)$
	2-truncated Witt vectors over $\Bbb F_2((t))$
	Algebraic closure of $F_2$
	field of $2$-adic numbers
	Field of algebraic numbers
	Field of constructible numbers
	Perfect non-Artinian ring
	Quasi-continuous ring that is not Ikeda-Nakayama

Name

$\mathbb Q[x,y]_{(x,y)}/(x^2-y^3)$

$C^\infty_0(\mathbb R)$: the ring of germs of smooth functions on $\mathbb R$ at $0$

$k[[x,y]]/(x^2,xy)$

ACCP ring with non-ACCP polynomial ring

Atomic domain with nonatomic polynomial ring

Clark's uniserial ring

Cohn's Schreier domain that isn't GCD

Custom Krull dimension valuation ring

Grams' atomic domain which doesn't satisfy ACCP

Mori but not Krull domain

Non-ACCP polynomial ring