Ring detail

Name: Finitely cogenerated, not semilocal ring.

Description: The ring is the trivial extension $T(\mathbb Z, \mathbb Z_{2^\infty})$ of the Prüfer $2$-group.

Notes: Krull dimension $1$.

Keywords trivial extension

Reference(s):

(Citation needed)

This ring has the following properties:

2-primal Abelian commutative connected CS (left) CS (right) Dedekind finite duo (left) duo (right) essential socle (left) essential socle (right) finite uniform dimension (left) finite uniform dimension (right) finitely cogenerated (left) finitely cogenerated (right) finitely generated socle (left) finitely generated socle (right) IBN Ikeda-Nakayama (left) Ikeda-Nakayama (right) lift/rad NI (nilpotents form an ideal) nil radical nilpotent radical nonzero socle (left) nonzero socle (right) Ore ring (left) Ore ring (right) orthogonally finite polynomial identity quasi-continuous (left) quasi-continuous (right) quasi-duo (left) quasi-duo (right) reversible semicommutative (SI condition, zero-insertive) simple socle (left) simple socle (right) stably finite strongly connected symmetric T-nilpotent radical (left) T-nilpotent radical (right) uniform (left) uniform (right)

The ring lacks the following properties:

$\pi$-regular $I_0$ Artinian (left) Artinian (right) Baer Bezout domain (left) Bezout domain (right) clean cogenerator ring (left) cogenerator ring (right) continuous (left) continuous (right) division ring domain dual (left) dual (right) exchange FI-injective (left) FI-injective (right) finite free ideal ring (left) free ideal ring (right) Frobenius fully prime fully semiprime hereditary (left) hereditary (right) Kasch (left) Kasch (right) local Noetherian (left) Noetherian (right) nonsingular (left) nonsingular (right) Ore domain (left) Ore domain (right) perfect (left) perfect (right) primary prime primitive (left) primitive (right) principal ideal domain (left) principal ideal domain (right) principal ideal ring (left) principal ideal ring (right) principally injective (left) principally injective (right) pseudo-Frobenius (left) pseudo-Frobenius (right) quasi-Frobenius reduced Rickart (left) Rickart (right) self-injective (left) self-injective (right) semi free ideal ring semihereditary (left) semihereditary (right) semilocal semiperfect semiprimary semiprime semiprimitive semiregular semisimple serial (left) serial (right) simple simple Artinian strongly $\pi$-regular strongly regular top regular top simple top simple Artinian unit regular V ring (left) V ring (right) valuation ring (left) valuation ring (right) von Neumann regular Zorn