On the strongly irreducible ideals of a duo ring

A proper ideal [Formula: see text] in a ring [Formula: see text] is called strongly irreducible (briefly, SI-ideal), if [Formula: see text] then [Formula: see text] or [Formula: see text], where [Formula: see text], [Formula: see text] are ideals of [Formula: see text]. We show that whenever in a local Noetherian duo domain, a power of its maximal ideal, [Formula: see text] say, is principal, then so is [Formula: see text]. It is also shown that whenever [Formula: see text] is an SI-ideal in a Noetherian duo ring [Formula: see text] where [Formula: see text] and [Formula: see text] with [Formula: see text], then [Formula: see text] is a domain. Finally in a Noetherian duo ring [Formula: see text], non-prime ideals with positive heights, which are SI-ideal, are characterized.