Ring $R_{ 70 }$

Bergman's exchange ring that isn't clean

Description:

Notes: Description is Šter's construction of the opposite ring of Bergman's original. It has been said that all examples of nonclean exchange rings so far are based on this one.

Expanded details page.

Keywords infinite matrix ring quotient ring subring

Reference(s):

D. Handelman. Perspectivity and cancellation in regular rings. (1977) @ Example 1
J. Ster. Corner rings of a clean ring need not be clean. (2012) @ Example 3.1
J. Ster. Lifting units in clean rings. (2013) @ Bergman's example
K. R. Goodearl. Von Neumann regular rings. (1991) @ Example 4.26, pp 46-48
T.-K. Lee, Z. Yi, and Y. Zhou. An example of Bergman's and the extension problem for clean rings. (2008) @ Bergman's example
V. P. Camillo and H.-P. Yu. Exchange rings, units and idempotents. (1994) @ p 4746

Symmetric properties

	Name
	$\pi$-regular
	$I_0$
	2-primal
	Abelian
	anti-automorphic
	Armendariz
	Baer
	Boolean
	clean
	commutative
	compressible
	countable
	Dedekind finite
	directly irreducible
	division ring
	domain
	exchange
	field
	finite
	Frobenius
	fully prime
	fully semiprime
	IBN
	IC ring
	involutive
	lift/rad
	local
	NI ring
	nil radical
	nilpotent radical
	orthogonally finite
	periodic
	polynomial identity
	potent
	primary
	prime
	quasi-Frobenius
	reduced
	reversible
	semi free ideal ring
	semicommutative
	semilocal
	semiperfect
	semiprimary
	semiprime
	semiprimitive
	semiregular
	semisimple
	simple
	simple Artinian
	stable range 1
	stably finite
	strongly $\pi$-regular
	strongly connected
	strongly regular
	symmetric
	top regular
	top simple
	top simple Artinian
	unit regular
	von Neumann regular
	weakly clean
	Zorn

Asymmetric properties

left	Name	right
	cogenerator ring
	CS
	dual
	FI-injective
	finitely pseudo-Frobenius
	hereditary
	Ikeda-Nakayama
	Kasch
	max ring
	primitive
	quasi-continuous
	semi-Noetherian
	simple-injective
	V ring
	ACC annihilator
	ACC principal
	Artinian
	Bezout domain
	cohopfian
	continuous
	DCC annihilator
	distributive
	duo
	finite uniform dimension
	finitely cogenerated
	finitely generated socle
	free ideal ring
	Goldie
	linearly compact
	McCoy
	Noetherian
	Ore domain
	PCI ring
	perfect
	principal ideal domain
	principal ideal ring
	pseudo-Frobenius
	quasi-duo
	self-injective
	serial
	simple socle
	UGP ring
	uniform
	uniserial domain
	uniserial ring
	Bezout
	coherent
	essential socle
	nonsingular
	nonzero socle
	Ore ring
	principally injective
	Rickart
	semi-Artinian
	semihereditary
	T-nilpotent radical

Legend

= has the property
= does not have the property
= information not in database

Name	Measure
composition length	left: $\infty$	right: $\infty$
weak global dimension	0

Name	Description
Jacobson radical	$\{0\}$
Left singular ideal	$\{0\}$
Right singular ideal	$\{0\}$