Ring detail

Name: Faith-Menal counterexample

Description: Let $D$ be a countable, existentially closed division ring over a field $F$. Let $S=D\otimes_F F(x)$. Let $R$ be the trivial extension $T(S,D)$ of $S$ by the $S$-module $D$$


Keywords rational polynomial ring tensor product trivial extension


  • Nicholson, W. Keith; Mohamed F. Yousif., Quasi-Frobenius Rings., Vol. 158. Cambridge University Press, (2003). Example 8.16 Pg 212

We don't know if the ring has or lacks the following properties:
$I_0$ 2-primal Abelian ACC annihilator (left) ACC principal (left) Baer Bezout (left) Bezout (right) clean cogenerator ring (right) coherent (left) cohopfian (left) cohopfian (right) commutative connected continuous (left) continuous (right) CS (left) CS (right) DCC annihilator (right) distributive (left) distributive (right) duo (left) duo (right) essential socle (left) essential socle (right) exchange FI-injective (left) finite uniform dimension (left) finitely cogenerated (left) finitely cogenerated (right) finitely generated socle (left) finitely pseudo-Frobenius (left) finitely pseudo-Frobenius (right) fully prime fully semiprime Goldie (left) hereditary (left) hereditary (right) Ikeda-Nakayama (left) Ikeda-Nakayama (right) Kasch (left) lift/rad local NI (nilpotents form an ideal) nil radical nilpotent radical Noetherian (left) nonsingular (left) nonsingular (right) nonzero socle (left) Ore ring (left) Ore ring (right) polynomial identity prime primitive (left) primitive (right) principal ideal ring (left) principal ideal ring (right) principally injective (left) quasi-continuous (left) quasi-continuous (right) quasi-duo (left) quasi-duo (right) reduced reversible Rickart (left) Rickart (right) semicommutative (SI condition, zero-insertive) semihereditary (left) semihereditary (right) semilocal semiperfect semiprime semiprimitive semiregular serial (left) serial (right) simple simple socle (left) simple socle (right) simple-injective (left) simple-injective (right) stable range 1 strongly connected symmetric T-nilpotent radical (left) T-nilpotent radical (right) top regular top simple top simple Artinian uniform (left) uniform (right) V ring (left) V ring (right) valuation ring (left) valuation ring (right) weakly clean Zorn