Ring $R_{ 145 }$

Trivial extension torch ring

Description:

Let $S$ be ring 144, and Q be its field of fractions, and let $M$ be a maximal ideal of $S$. The ring is the trivial extension $T(S, Q/S_{M})$

Reference(s):

T. S. Shores and R. Wiegand. Rings whose finitely generated modules are direct sums of cyclics. (1974) @ Example 4.5 p 166

Known Properties

	Name
	ACC annihilator
	almost maximal ring
	catenary
	coherent
	cohopfian
	CS
	DCC annihilator
	essential socle
	FI-injective
	finite uniform dimension
	finitely cogenerated
	finitely generated socle
	Goldie
	Ikeda-Nakayama
	J-0
	J-1
	J-2
	Kasch
	nonzero socle
	principally injective
	quasi-continuous
	semi-Noetherian
	simple socle
	simple-injective
	uniform
	universally catenary
	universally Japanese
	$\pi$-regular
	$h$-local domain
	$I_0$
	ACC principal
	algebraically closed field
	almost Dedekind domain
	almost maximal domain
	almost maximal valuation ring
	analytically normal
	analytically unramified
	Archimedean field
	Artinian
	atomic domain
	Baer
	Bezout domain
	Boolean
	characteristic 0 field
	clean
	cogenerator ring
	Cohen-Macaulay
	complete discrete valuation ring
	complete local
	continuous
	countable
	Dedekind domain
	discrete valuation ring
	division ring
	domain
	dual
	Euclidean domain
	Euclidean field
	excellent
	exchange
	field
	finite
	free ideal ring
	Frobenius
	fully prime
	fully semiprime
	GCD domain
	Goldman domain
	Gorenstein
	Grothendieck
	Henselian local
	hereditary
	Jacobson
	Krull domain
	lift/rad
	linearly compact
	local
	local complete intersection
	max ring
	maximal ring
	maximal valuation ring
	Mori domain
	N-1
	N-2
	Nagata
	nil radical
	nilpotent radical
	Noetherian
	nonsingular
	normal
	normal domain
	ordered field
	Ore domain
	PCI ring
	perfect
	perfect field
	periodic
	potent
	primary
	prime
	primitive
	principal ideal domain
	principal ideal ring
	Prufer domain
	pseudo-Frobenius
	Pythagorean field
	quadratically closed field
	quasi-excellent
	quasi-Frobenius
	rad-nil
	reduced
	regular
	regular local
	Rickart
	Schreier domain
	self-injective
	semi free ideal ring
	semi-Artinian
	semihereditary
	semiperfect
	semiprimary
	semiprime
	semiprimitive
	semiregular
	semisimple
	serial
	simple
	simple Artinian
	strongly $\pi$-regular
	strongly regular
	T-nilpotent radical
	top simple
	top simple Artinian
	unique factorization domain
	uniserial domain
	uniserial ring
	unit regular
	V ring
	valuation domain
	valuation ring
	von Neumann regular
	weakly clean
	Zorn
	2-primal
	?-ring
	Abelian
	anti-automorphic
	arithmetical
	Armendariz
	Bezout
	commutative
	compressible
	Dedekind finite
	directly irreducible
	distributive
	duo
	FGC
	finitely pseudo-Frobenius
	IBN
	IC ring
	involutive
	McCoy
	NI ring
	Ore ring
	orthogonally finite
	polynomial identity
	quasi-duo
	reversible
	semicommutative
	semilocal
	stable range 1
	stably finite
	strongly connected
	symmetric
	top regular
	torch
	UGP ring

Legend

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

(Nothing was retrieved.)

Name	Description
Idempotents	$\{0,1\}$