Arapović rings and modules

If [Formula: see text] is a strongly Hopfian ring such that the intersection of the kernels of all ring homomorphism [Formula: see text] (where [Formula: see text] ranges over the minimal prime ideals of [Formula: see text]) is zero, then [Formula: see text] is embeddable in a zero-dimensional ring. Motivated by this, we consider rings in which the above intersection is zero, called Arapović rings. We investigate properties of Arapović rings. We prove that a ring [Formula: see text] is Arapović if and only if every ideal of [Formula: see text] with nonzero annihilator is contained in a minimal prime ideal of [Formula: see text]. We prove that [Formula: see text] is an Arapović ring if and only if [Formula: see text] is an Arapović ring. For a chained ring [Formula: see text] with nilradical [Formula: see text], we prove that [Formula: see text] is an Arapović ring if and only if the zero ideal of [Formula: see text] is primary and (either [Formula: see text] is SFT or [Formula: see text] has zero annihilator). We study Gilmer rings, that is rings in which the intersection of all primary ideals is zero. We illustrate results with examples.