On σ-primeness and σ-semiprimeness in rings with involution

This study introduces the structure of $\left\vert \mathcal{S}_{R}^{\sigma}\right\vert $-$\sigma$-semiprime ring, $\left\vert \mathcal{S}_{R}^{\sigma}\right\vert $-$\sigma$-prime ring, and source of $\sigma$-primeness , which have not been previously explored, and presents new results. We define a subset $$ P_{R_{\sigma}} = \bigcap_{a\in R} S_{R_{\sigma}}^{a} $$ of ring $R$ as $S_{R}^{\sigma}=\left\{ \left. a\in R\right\vert aRa=aR{\sigma}(a)=(0)\right\} $, where $\sigma$ is an involution is referred to as the source of $\sigma$-primeness of $R$. Additionally, we have established some relationships between the prime radical $\beta(R)$ and $\mathcal{S}_{R}^{\sigma}$.