# Ring detail

## Name: Noetherian domain that is not N-1

Description: Let $x_i$ be countably many indeterminates, and $T=\mathbb Q[\{x_i^2\mid i \in \mathbb N\}\cup\{x_i^3\in\mathbb N\}]$. Let $S_0$ be the multiplicative set that is the intersection of complements of $(x_i^2, x_i^3)$ in $\mathbb Q[\{x_i\mid i \in \mathbb N\}]$, and $S=S_0\cap T$. The desired ring is the localization TS^{-1}$Notes: not integrally closed in its field of fractions Keywords localization polynomial ring Reference(s): • (Citation needed) • Known Properties Name$I_0\pi$-regular 2-primal ACC annihilator ACC principal Abelian Archimedean field Armendariz Artinian Baer Bezout Bezout domain Boolean CS Cohen-Macaulay DCC annihilator Dedekind domain Dedekind finite Euclidean domain Euclidean field FI-injective Frobenius GCD domain Goldie Goldman domain Gorenstein Grothendieck Henselian local IBN Ikeda-Nakayama J-0 J-1 J-2 Jacobson (Hilbert) Japanese (N-2) Kasch Krull domain McCoy Mori domain N-1 NI (nilpotents form an ideal) Nagata Noetherian Ore domain Ore ring PCI ring Prufer domain Pythagorean field Rickart Schreier domain T-nilpotent radical V ring Zorn algebraically closed field almost Dedekind domain analytically normal analytically unramified atomic domain catenary characteristic 0 field clean cogenerator ring coherent cohopfian commutative complete local compressible connected continuous countable distributive division ring domain dual duo essential socle excellent exchange field finite finite uniform dimension finitely cogenerated finitely generated socle finitely pseudo-Frobenius free ideal ring fully prime fully semiprime hereditary lift/rad local local complete intersection max ring nil radical nilpotent radical nonsingular nonzero socle normal normal domain ordered field orthogonally finite perfect perfect field periodic polynomial identity potent primary prime primitive principal ideal domain principal ideal ring principally injective pseudo-Frobenius quadratically closed field quasi-Frobenius quasi-continuous quasi-duo quasi-excellent rad-nil reduced regular regular local reversible self-injective semi free ideal ring semicommutative (SI condition, zero-insertive) semihereditary semilocal semiperfect semiprimary semiprime semiprimitive semiregular semisimple serial simple simple Artinian simple socle simple-injective stable range 1 stably finite strongly$\pi\$-regular
strongly connected
strongly regular
symmetric
top regular
top simple
top simple Artinian
uniform
unique factorization domain
uniserial ring
unit regular
universally Japanese
universally catenary
valuation domain
von Neumann regular
weakly clean
Legend
• = has the property
• = does not have the property
• = information not in database

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