▁regular ▁rings , ▁and ▁an ▁ ▁isomorphism ▁theorem ▁for ▁rings ▁establish ing ▁that ▁R / K <0xE2> <0x89> <0x85> ( R / J )/ ( J / K ). ▁In ▁particular ▁if ▁A ▁is ▁any ▁ideal ▁in ▁R ▁the ▁ideal ▁A + J ▁is ▁quot ient ▁von ▁Ne umann ▁regular ▁if ▁J ▁is . ▁ ▁Ex amples ▁ ▁Every ▁proper ▁ideal ▁of ▁a ▁von ▁Ne umann ▁regular ▁ring ▁is ▁quot ient ▁von ▁Ne umann ▁regular . ▁ ▁Any ▁maximal ▁ideal ▁in ▁a ▁commut ative ▁ring ▁is ▁a ▁quot ient ▁von ▁Ne umann ▁regular ▁ideal ▁since ▁R / M ▁is ▁a ▁field . ▁This ▁is ▁not ▁true ▁in ▁general ▁because ▁for ▁non comm ut ative ▁rings ▁R / M ▁may ▁only ▁be ▁a ▁simple ▁ring , ▁and ▁may ▁not ▁be ▁von ▁Ne umann ▁regular . ▁ ▁Let ▁R ▁be ▁a ▁local ▁ring ▁which ▁is ▁not ▁a ▁division ▁ring , ▁and ▁with ▁maximal ▁right ▁ideal ▁M ▁. ▁Then ▁M ▁is ▁a ▁quot ient ▁von ▁Ne umann ▁regular ▁ideal , ▁since ▁R / M ▁is ▁a ▁division ▁ring , ▁but ▁R ▁is ▁not ▁a ▁von ▁Ne umann ▁regular ▁ring . ▁ ▁More ▁generally ▁in ▁any ▁sem il ocal ▁ring ▁the ▁Jacob son ▁radical ▁J ▁is ▁quot ient ▁von ▁Ne umann ▁regular , ▁since ▁R / J '' ▁is ▁a ▁sem is im ple ▁ring , ▁hence ▁a ▁von ▁Ne umann ▁regular ▁ring . ▁ ▁References ▁▁▁▁▁ ▁Category : R ing ▁theory ▁Category : I de als <0x0A> </s> ▁Ma q so od ▁Me hm ood ▁Ali