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.