Ring $R_{ 119 }$

$\mathbb Q\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra

Description:

The ring is the quotient $\mathbb Q\langle x,y \rangle/(xy-1)$.

Notes: Minimum nonzero ideal is the left/right socle, generated by the $E_{i,j}$

Keywords free algebra quotient ring

Reference(s):

M. C. Iovanov and A. Sistko. On the Toeplitz-Jacobson algebra and direct finiteness. (2016) @ (whole article)
T. Lam. A first course in noncommutative rings. (2013) @ Section 11 p 184
V. Bavula. The algebra of one-sided inverses of a polynomial algebra. (2010) @ (whole article)

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
	subdirectly irreducible
	symmetric
	top regular
	top simple
	top simple Artinian
	unit regular
	von Neumann regular
	weakly clean
	Zorn

Asymmetric properties

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

Legend

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

Name	Measure
Krull dimension (classical)	2

Name	Description
Jacobson radical	$\{0\}$
Left socle	$FM_\omega(\mathbb Q)$
Right socle	$FM_\omega(\mathbb Q)$