A Generalization of Source of Semiprimeness
Abstract
This 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 $\mathcal{L}_{R}^{n}(I)$.
Keywords
Source of semiprimeness, semiprime rings, semiprime ideals, prime rings, prime ideals
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