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Let $R$ be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of $n$-ideals and the class of $(2,n)$-ideals. A proper ideal $I$ of $R$ is said to be a quasi $n$-ideal if $\sqrt{I}$ is an $n$-ideal of $R.$ Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the $n$-ideals, the quasi primary ideals, the $(2,n)$-ideals and the $pr$-ideals. Moreover, we use the quasi $n$-ideals to characterize some...