Rings and radicals related to n-primariness

Abstract

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].