Camcı, Didem KaralarlıoğluYeşil, DidemMekera, RasieCamcı, Çetin2025-05-292025-05-2920242149-1402https://doi.org/10.53570/jnt.1581076https://hdl.handle.net/20.500.12428/31613This paper characterizes the semigroup ideal $\mathcal{L}_{R}^{n}(I)$ of a ring $R$, where $I$ is an ideal of $R$, defined by $\mathcal{L}_{R}^{0}(I)=I$ and $\mathcal{L}_{R}^{n}(I)=\{a\in R \mid aRa\subseteq \mathcal{L}_{R}^{n-1}(I)\}$, for all $n\in \mathbb{Z}^+$, the set of all the positive integers. Moreover, it studies the basic properties of the set $\mathcal{L}_{R}^{n}(I)$ and defines $n$-prime ideals, $n$-semiprime ideals, $n$-prime rings, and $n$-semiprime rings. This study also investigates relationships between the sets $\mathcal{L}_{R}(I)$ and $\mathcal{L}_{R}^{n}(I)$ and exemplifies some of the related properties. It obtains the main results concerning prime rings and prime ideals by the properties of the set $\mathcal{L}_{R}^{n}(I)$.eninfo:eu-repo/semantics/openAccessSource of semiprimenesssemiprime ringssemiprime idealsprime ringsprime idealsResearch Article49626810.53570/jnt.1581076