Rings and radicals related to n-primariness

This paper concerns ring properties which are induced from the structure of the powers of prime ideals. An ideal Formula: see text of a ring Formula: see text is called Formula: see text-primary (respectively, Formula: see text-primary) provided that Formula: see text for ideals Formula: see text of Formula: see text implies that Formula: see text or Formula: see text is nil of index Formula: see text (respectively, Formula: see text or Formula: see text is nil) in Formula: see text, where Formula: see text. It is proved that for a proper ideal Formula: see text of a principal ideal domain Formula: see text, Formula: see text is Formula: see text-primary if and only if Formula: see text is of the form Formula: see text for some prime element Formula: see text and Formula: see text if and only if Formula: see text is Formula: see text-primary, through which we study the structure of matrices over principal ideal domains. We prove that for a Formula: see text-primary ideal Formula: see text of a ring Formula: see text, Formula: see text is prime when the Wedderburn radical of Formula: see text is zero. In addition we provide a method of constructing strictly descending chain of Formula: see text-primary radicals from any domain, where the Formula: see text-primary radical of a ring Formula: see text means the intersection of all the Formula: see text-primary ideals of Formula: see text.