▁. ▁▁ ▁In ▁commut ative ▁algebra ▁a ▁regular ▁ideal ▁refers ▁to ▁an ▁ideal ▁containing ▁a ▁non - zero ▁divis or . ▁ ▁This ▁article ▁will ▁use ▁" regular ▁element ▁ideal " ▁to ▁help ▁distinguish ▁this ▁type ▁of ▁ideal . ▁ ▁A ▁two - s ided ▁ideal ▁ ▁of ▁a ▁ring ▁R ▁can ▁also ▁be ▁called ▁a ▁( von ▁Ne umann ) ▁regular ▁ideal ▁if ▁for ▁each ▁element ▁x ▁of ▁ ▁there ▁exists ▁a ▁y ▁in ▁ ▁such ▁that ▁x y x = x . ▁▁▁ ▁Finally , ▁regular ▁ideal ▁has ▁been ▁used ▁to ▁refer ▁to ▁an ▁ideal ▁J ▁of ▁a ▁ring ▁R ▁such ▁that ▁the ▁quot ient ▁ring ▁R / J ▁is ▁von ▁Ne umann ▁regular ▁ring .< ref > B ur ton , ▁D . M . ▁( 1 9 7 0 ) ▁`` A ▁first ▁course ▁in ▁rings ▁and ▁ide als . ▁Add ison - W es ley . ▁Reading , ▁Massachusetts ▁. </ ref > ▁This ▁article ▁will ▁use ▁" quot ient ▁von ▁Ne umann ▁regular " ▁to ▁refer ▁to ▁this ▁type ▁of ▁regular ▁ideal . ▁ ▁Since ▁the ▁ad jective ▁regular ▁has ▁been ▁over loaded , ▁this ▁article ▁adopt s ▁the ▁alternative ▁ad ject ives ▁mod ular , ▁regular ▁element , ▁von ▁Ne umann ▁regular , ▁and ▁quot ient ▁von ▁Ne umann ▁regular ▁to ▁distinguish ▁between ▁concepts . ▁ ▁Properties ▁and ▁examples ▁ ▁Mod ular ▁ide als ▁The ▁notion ▁of ▁mod ular ▁ide als ▁perm its ▁the ▁general ization ▁of ▁various ▁character izations ▁of ▁ide als ▁in ▁a ▁un