▁ideal , ▁namely ▁{ 0 }. ▁Furthermore , ▁every ▁ring ▁has ▁a ▁maximal ▁von ▁Ne umann ▁regular ▁ideal ▁containing ▁all ▁other ▁von ▁Ne umann ▁regular ▁ide als , ▁and ▁this ▁ideal ▁is ▁given ▁by ▁. ▁ ▁Ex amples ▁ ▁As ▁noted ▁above , ▁every ▁ideal ▁of ▁a ▁von ▁Ne umann ▁regular ▁ring ▁is ▁a ▁von ▁Ne umann ▁regular ▁ideal . ▁ ▁It ▁is ▁well ▁known ▁that ▁a ▁local ▁ring ▁which ▁is ▁also ▁a ▁von ▁Ne umann ▁regular ▁ring ▁is ▁a ▁division ▁ring . ▁Let ▁R ▁Be ▁a ▁local ▁ring ▁which ▁is ▁not ▁a ▁division ▁ring , ▁and ▁denote ▁the ▁unique ▁maximal ▁right ▁ideal ▁by ▁J . ▁Then ▁R ▁cannot ▁be ▁von ▁Ne umann ▁regular , ▁but ▁R / J , ▁being ▁a ▁division ▁ring , ▁is ▁a ▁von ▁Ne umann ▁regular ▁ring . ▁Con sequently , ▁J ▁cannot ▁be ▁a ▁von ▁Ne umann ▁regular ▁ideal , ▁even ▁though ▁it ▁is ▁maximal . ▁ ▁A ▁simple ▁domain ▁which ▁is ▁not ▁a ▁division ▁ring ▁has ▁the ▁minimum ▁possible ▁number ▁of ▁von ▁Ne umann ▁regular ▁ide als : ▁only ▁the ▁{ 0 } ▁ideal . ▁ ▁Qu ot ient ▁von ▁Ne umann ▁regular ▁ide als ▁If ▁J ▁and ▁K ▁are ▁quot ient ▁von ▁Ne umann ▁regular ▁ide als , ▁then ▁so ▁is ▁J ∩ K . ▁ ▁If ▁J ⊆ K ▁are ▁proper ▁ide als ▁of ▁R ▁and ▁J ▁is ▁quot ient ▁von ▁Ne umann ▁regular , ▁then ▁so ▁is ▁K . ▁This ▁is ▁because ▁quot ients ▁of ▁R / J ▁are ▁all ▁von ▁Ne umann