Almost Condensed Domains

As an extension of the class of half condensed domains introduced by D.D. Anderson and Dumitrescu, we introduce and study the class of almost condensed domains. An integral domain D is almost condensed if whenever 0 ̸= z ∈ IJ with I, J ideals of D, there exist I ′, J ′ ideals of D such that I ′ ⊆ Iw, J ′ ⊆ Jw and zD = (I ′J )w. In 1983, D.F. Anderson and D.E. Dobbs [13] called an integral domain D condensed if for each pair of ideals I, J ofD, IJ = {ij | i ∈ I, j ∈ J}. A Bézout domain is condensed, cf. [13, Corollary 2.2] and an integrally closed condensed domain is Bézout, cf. [11, Main Theorem]. In 2004, D.D. Anderson and T. Dumitrescu [6] called an integral domain D a half condensed (HC) domain if whenever 0 ̸= z ∈ IJ with I, J ideals of D, there exist I ′, J ′ (invertible) ideals of D such that I ′ ⊆ I, J ′ ⊆ J and zD = I ′J ′. In this paper we study the following extension of the concept of HC domain. We call an integral domain D an almost condensed (AC) domain if whenever 0 ̸= z ∈ IJ with I, J ideals of D, there exist I ′, J ′ (w-invertible) ideals of D such that I ′ ⊆ Iw, J ′ ⊆ Jw and zD = (I ′J )w (the definition of the w-closure of an ideal as well as other basic facts are recalled below). The following implications hold.