ital ▁ring ▁to ▁non - un ital ▁settings . ▁ ▁A ▁two - s ided ▁ideal ▁ ▁is ▁mod ular ▁if ▁and ▁only ▁if ▁ ▁is ▁un ital . ▁In ▁a ▁un ital ▁ring , ▁every ▁ideal ▁is ▁mod ular ▁since ▁choosing ▁e = 1 ▁works ▁for ▁any ▁right ▁ideal . ▁So , ▁the ▁notion ▁is ▁more ▁interesting ▁for ▁non - un ital ▁rings ▁such ▁as ▁Ban ach ▁al gebras . ▁From ▁the ▁definition ▁it ▁is ▁easy ▁to ▁see ▁that ▁an ▁ideal ▁containing ▁a ▁mod ular ▁ideal ▁is ▁itself ▁mod ular . ▁ ▁Some what ▁surprising ly , ▁it ▁is ▁possible ▁to ▁prove ▁that ▁even ▁in ▁rings ▁without ▁identity , ▁a ▁mod ular ▁right ▁ideal ▁is ▁contained ▁in ▁a ▁maximal ▁right ▁ideal . ▁However , ▁it ▁is ▁possible ▁for ▁a ▁ring ▁without ▁identity ▁to ▁lack ▁mod ular ▁right ▁ide als ▁entirely . ▁ ▁The ▁intersection ▁of ▁all ▁maximal ▁right ▁ide als ▁which ▁are ▁mod ular ▁is ▁the ▁Jacob son ▁radical . ▁ ▁Ex amples ▁ ▁In ▁the ▁non - un ital ▁ring ▁of ▁even ▁integers , ▁( 6 ) ▁is ▁regular ▁() ▁while ▁( 4 ) ▁is ▁not . ▁ ▁Let ▁M ▁be ▁a ▁simple ▁right ▁A - module . ▁If ▁x ▁is ▁a ▁non zero ▁element ▁in ▁M , ▁then ▁the ▁anni h il ator ▁of ▁x ▁is ▁a ▁regular ▁maximal ▁right ▁ideal ▁in ▁A . ▁ ▁If ▁A ▁is ▁a ▁ring ▁without ▁maximal ▁right ▁ide als , ▁then ▁A ▁cannot ▁have ▁even ▁a ▁single ▁mod ular ▁right ▁ideal . ▁ ▁Reg