The notion of a left {\rm(}resp., right{\rm)} ${\mathcal N}_{\mathcal I}$-ideal is introduced, and related properties are investigated. Characterizations of a left {\rm(}resp., right{\rm)} ${\mathcal N}_{\mathcal I}$-ideal are considered. Translations of a left {\rm(}resp., right{\rm)} ${\mathcal N}_{\mathcal I}$-ideal are studied. We show that the homomorphic image (preimage) of a left {\rm(}resp., right{\rm)} ${\mathcal N}_{\mathcal I}$-ideal is a left {\rm(}resp., right{\rm)} ${\mathcal N}_{\mathcal I}$-ideal. The notion of retrenched left {\rm(}resp., right{\rm)} ${\mathcal N}_{\mathcal I}$-ideals is introduced, and their properties are investigated.