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 NI (nilpotents form an ideal) 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 $I_0$ Abelian Artinian (left) Artinian (right) Bezout domain (left) Bezout domain (right) clean cogenerator ring (left) cogenerator ring (right) commutative continuous (left) continuous (right) distributive (left) distributive (right) division ring domain dual (left) dual (right) duo (left) duo (right) exchange FI-injective (left) FI-injective (right) finite free ideal ring (left) free ideal ring (right) Frobenius fully prime fully semiprime 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 reversible self-injective (left) self-injective (right) semi free ideal ring semicommutative (SI condition, zero-insertive) semilocal semiperfect semiprimary semiprime semiprimitive semiregular semisimple serial (left) serial (right) simple simple Artinian strongly $\pi$-regular strongly connected 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) connected CS (left) CS (right) 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-continuous (left) quasi-continuous (right) quasi-duo (left) quasi-duo (right) 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 weakly clean