Commutative rings in which zero-components of essential primes are essential

Abstract

For a prime ideal [Formula: see text] of a commutative ring [Formula: see text] with identity, we denote (as usual) by [Formula: see text] its zero-component; that is, the set of members of [Formula: see text] that are annihilated by nonmembers of [Formula: see text]. We study rings in which [Formula: see text] is an essential ideal, whenever [Formula: see text] is an essential prime ideal. We characterize them in terms of the lattices (which are, in fact, complete Heyting algebras) of their radical ideals. We prove that the classical ring of quotients of any ring of this kind is itself of this kind. We show that direct products of rings of this kind are themselves of this kind. We observe that the ring of real-valued continuous functions on a Tychonoff space is of this kind precisely when the underlying set of the space is infinite. Replacing [Formula: see text] with the pure part of [Formula: see text], we obtain a formally stronger variant which is still characterizable in terms of the lattices of radical ideals.