umann ▁regular ▁ring ▁if ▁and ▁only ▁if ▁R ▁is ▁a ▁von ▁Ne umann ▁regular ▁ideal . ▁The ▁following ▁statement ▁is ▁a ▁relevant ▁lemma ▁for ▁von ▁Ne umann ▁regular ▁ide als : ▁ ▁Lemma : ▁For ▁a ▁ring ▁R ▁and ▁proper ▁ideal ▁J ▁containing ▁an ▁element ▁a , ▁there ▁exists ▁and ▁element ▁y ▁in ▁J ▁such ▁that ▁a = aya ▁if ▁and ▁only ▁if ▁there ▁exists ▁an ▁element ▁r ▁in ▁R ▁such ▁that ▁a = ara . ▁Proof : ▁The ▁" only ▁if " ▁direction ▁is ▁a ▁t aut ology . ▁For ▁the ▁" if " ▁direction , ▁we ▁have ▁a = ara = ar ara . ▁Since ▁a ▁is ▁in ▁J , ▁so ▁is ▁r ar , ▁and ▁so ▁by ▁setting ▁y = rar ▁we ▁have ▁the ▁conclusion . ▁ ▁As ▁a ▁consequence ▁of ▁this ▁lemma , ▁it ▁is ▁apparent ▁that ▁every ▁ideal ▁of ▁a ▁von ▁Ne umann ▁regular ▁ring ▁is ▁a ▁von ▁Ne umann ▁regular ▁ideal . ▁Another ▁consequence ▁is ▁that ▁if ▁J ▁and ▁K ▁are ▁two ▁ide als ▁of ▁R ▁such ▁that ▁J ⊆ K ▁and ▁K ▁is ▁a ▁von ▁Ne umann ▁regular ▁ideal , ▁then ▁J ▁is ▁also ▁a ▁von ▁Ne umann ▁regular ▁ideal . ▁ ▁If ▁J ▁and ▁K ▁are ▁two ▁ide als ▁of ▁R , ▁then ▁K ▁is ▁von ▁Ne umann ▁regular ▁if ▁and ▁only ▁if ▁both ▁J ▁is ▁a ▁von ▁Ne umann ▁regular ▁ideal ▁and ▁K / J ▁is ▁a ▁von ▁Ne umann ▁regular ▁ring . ▁ ▁Every ▁ring ▁has ▁at ▁least ▁one ▁von ▁Ne umann ▁regular