On σ-primeness and σ-semiprimeness in rings with involution - Barış Albayrak; Didem Karalarlıoğlu Camcı; Didem Yeşil (Journal of New Results in Science)
This study introduces the structure of $\left\vert\mathcal{S}_{R}^{σ}\right\vert $-$σ$-semiprime ring, $\left\vert \mathcal{S}_{R}^{σ}\right\vert $-$σ$-prime ring, and source of $σ$-primeness , which have not been previously explored, and presents new results. We define a subset $$P_{R_{σ}} = \bigcap_{a\in R} S_{R_{σ}}^{a}$$ of ring $R$ as $S_{R}^{σ}=\left\{ \left. a\in R\right\vert aRa=aRσ(a)=(0)\right\} $, where $σ$ is an involution is referred to as the source of $σ$-primeness of $R$. Additionally, we have established some relationships between the prime radical$β(R)$ and $\mathcal{S}_{R}^{σ}$.