Let R be an integral domain. An element u of the quotient field of R is said to be pseudo-integral over R if uIv ⊆ Iv for some nonzero finitely generated ideal I of R. The set of all pseudo-integral elements forms an integrally closed (but not necessarily pseudo-integrally closed) overling R ofR. It is shown that , where X is a family of indeterminates; pseudo-integrality is analyzed in rings of the form D + M; and an example is given to show that pseudo-integrality does not behave well with respect to localization.

A famous theorem of Krull's is that the integral closure of an integral domain D is the intersection of the valuation domains that contain D. An example is given to show that the same result need not hold for the integral closure of a ring with zero divisors.

We give a simple construction of non-Noetherian Hilbert domains whose maximal ideals are all finitely generated. Such domains we call unruly Hilbert domains.

A condensed domain is an integral domain such that IJ = {xy : x ∊ I, y ∊ J } holds for each pair I, J of ideals. We prove that, under suitable conditions, a subring of a discrete valuation ring is condensed if and only if it contains an element of value 2. We also define the concept strongly condensed.

The following two theorems are proved. If R is an Archimedean conducive integral domain, then R is quasilocal and dim(R) ≤1. If each overring of an integral domain R has ascending chain condition on divisorial ideals, then the integral closure of R is a Dedekind domain. Both theorems sharpen results already known in the Noetherian case. The second theorem leads to a strengthened converse of the Krull-Akizuki Theorem. We also investigate the effect of restricting the hypothesis in the second theorem to the proper overrings of R.

It is proved that if u is an element of a faithful algebra over a commutative ring R, then u satisfies a polynomial over R which has unit content if and only if the extension R ⊂ R[u] has the imcomparability property. Applications include new proofs of results of Gilmer-Hoffmann and Papick, as well as a characterization of the P-extensions introduced by Gilmer and Hoffmann.

All rings have an identity, all homomorphisms map identities to identities, all homomorphisms on algebras over fields are algebra homomorphisms. A ring R is a quotient-embeddable ring (a QE-ring) if for any proper ideal a of R there is an endomorphism of R whose kernel is the ideal a. A QE-ring U is a receptor of R if for any proper ideal a of R there is a homomorphism from R to U whose kernel is the ideal a.

Theorem. A ring R has a receptor if and only if it is a K-algebra over some field K contained in the center of R. If R is a commutative K-algebra of this type, then it has a commutative receptor.