Structure and topological properties of σ-radicals and σ-prime ideals induced by automorphism σ

We investigate the structure and topological properties of the Formula: see text-prime radical and strongly Formula: see text-prime ideals in a ring Formula: see text with an automorphism Formula: see text. Motivated by the characterization of the Formula: see text-prime radical given by Cheon et al., we provide a simpler description and apply it to analyze the relationships among various classes of Formula: see text-radicals. Building upon the topological approaches developed by Shin and others, we introduce a topology on the set of strongly Formula: see text-prime ideals and examine its compactness and separation properties. In particular, we show that Formula: see text coincides with the set of all Formula: see text-sequentially Formula: see text-nilpotent elements; that Formula: see text is Formula: see text-rigid if and only if every minimal strongly Formula: see text-prime ideal is completely Formula: see text-prime; and that Formula: see text and Formula: see text are compact spaces with further separation conditions characterized. These results generalize known radical-theoretic and topological properties from classical and Formula: see text-structured perspectives.