Source of semiprimeness of $\ast$-prime rings

Abstract

This study constructs a structure $S_{R}^{\ast}$ that had never been studied before and obtained new results by defining a subset $S_{R}^{\ast}$ of $R$ as$S_{R}^{\ast}=\left\{ \left. a\in R\right\vert aRa=aRa^{\ast}=(0)\right\} $ where $\ast$ is an involution and it is called as the source of $\ast$-semiprimeness of $R$. Moreover, it investigates some properties of the subset $S_{R}^{\ast}$ in any ring $R$. Additionally, the relation between the prime radical, which provides the opportunity to work on prime rings, has been studied in many ways, and the set $\mathcal{S}_{R}^{\sigma}$, the motivation of this study, is provided. Furthermore, it is proved that $\mathcal{S}_{R}^{\sigma}=\{0\}$ in the case where the ring $R$ is a reduced ($\sigma$-semiprime) ring and $f(\mathcal{S}_{R}^{\sigma})=\mathcal{S}_{f(R)}^{\sigma}$ under certain conditions for a ring homomorphism $f$. Besides, it is presented that for the idempotent element $e$, the inclusion $e\mathcal{S}_{R}^{\sigma}e\subseteq\mathcal{S}_{eRe}^{\sigma}$ is provided, and for the right ideal (ideal) $I$ of the ring $R$, $\mathcal{S}_{R}^{\sigma}(I)$ is a left semigroup ideal (semigroup ideal) of the multiplicative semigroup $R$. In addition, it is analyzed that the set $\mathcal{S}_{R}^{\sigma}$ is contained by the intersection of all semiprime ideals of the ring $R$.

Keywords

• involution
• ∗-prime ring
• ∗-semiprime ring
• source of semiprimeness