Ring detail

Name: Non-symmetric $2$-primal ring

Description: Consider the semigroup $S=\{a,b\}$ with $a^2=ab=a$,$b^2=ba=b$. Make the semigroup ring $T=F_2[S]$. Make the Dorroh extension $T'=\mathbb Z(+)T$. $T'$ is the ring.

Notes:

Keywords Dorroh extension

Reference(s):

  • Greg Marks, Reversible And Symmetric Rings, Journal Of Pure And Applied Algebra Volume 174, Issue 3, Pp 311–318, (2002). Example 2 P 313

This ring has the following properties:
2-primal ACC annihilator (left) ACC annihilator (right) ACC principal (left) ACC principal (right) coherent (left) coherent (right) DCC annihilator (left) DCC annihilator (right) Dedekind finite finite uniform dimension (left) finite uniform dimension (right) finitely generated socle (left) finitely generated socle (right) Goldie (left) Goldie (right) IBN lift/rad nil radical nilpotent radical Noetherian (left) Noetherian (right) orthogonally finite stably finite T-nilpotent radical (left) T-nilpotent radical (right)
The ring lacks the following properties:
$\pi$-regular Abelian Artinian (left) Artinian (right) Bezout domain (left) Bezout domain (right) clean cogenerator ring (left) cogenerator ring (right) commutative connected continuous (left) continuous (right) distributive (left) distributive (right) division ring domain dual (left) dual (right) duo (right) exchange FI-injective (left) FI-injective (right) finite free ideal ring (left) free ideal ring (right) Frobenius fully prime fully semiprime I_0 Kasch (left) Kasch (right) local Ore domain (left) Ore domain (right) perfect (left) perfect (right) primary prime primitive (left) primitive (right) principal ideal domain (left) principal ideal domain (right) principally injective (left) principally injective (right) pseudo-Frobenius (left) pseudo-Frobenius (right) quasi-Frobenius reduced self-injective (left) self-injective (right) semi free ideal ring semilocal semiperfect semiprimary semiprime semiprimitive semiregular semisimple serial (left) serial (right) simple simple Artinian strongly $\pi$-regular strongly regular symmetric top regular top simple Artinian uniform (left) uniform (right) unit regular V ring (left) V ring (right) valuation ring (left) valuation ring (right) von Neumann regular Zorn
We don't know if the ring has or lacks the following properties:
Baer Bezout (left) Bezout (right) cohopfian (left) cohopfian (right) duo (left) essential socle (left) essential socle (right) finitely cogenerated (left) finitely cogenerated (right) finitely pseudo-Frobenius (left) finitely pseudo-Frobenius (right) hereditary (left) hereditary (right) Ikeda-Nakayama (left) Ikeda-Nakayama (right) nonsingular (left) nonsingular (right) nonzero socle (left) nonzero socle (right) Ore ring (left) Ore ring (right) polynomial identity principal ideal ring (left) principal ideal ring (right) quasi-duo (left) quasi-duo (right) reversible Rickart (left) Rickart (right) semihereditary (left) semihereditary (right) simple socle (left) simple socle (right) simple-injective (left) simple-injective (right) stable range 1 top simple