Universally catenarian and going-down pairs of rings

For a ring extension is called a universally catenarian pair if every domain , is universally catenarian. When R is a field it is shown that the only universally catenarian pairs are those satisfying . For several satisfactory results are given. The second purpose of this paper is to study going-down pairs (Definition 5.1). We characterize these pairs of rings and we establish a relationship between universally catenarian, going-down and residually algebraic pairs. Received: 1 July 1999; in final form: 5 June 2000 / Published online: 17 May 2001     

A characterization of going-down-ring pairs of commutative rings

SoientA ⊆B des anneaux (commutaifs et unitaires). On dit que (A,B) est une paire d’anneaux de going-down siD est un anneau de going-down pour tout anneauD tel queA ⊆D ⊆B. On preuve que (A,B) est une paire d’anneaux de going-down si et seulement siA[b ₁,b ₂] est un anneau de going-down pour toutb ₁,b ₂ εB.     

A Generalization of Divided Domains and Its Connection to Weak Baer Going-Down Rings

Many results on going-down domains and divided domains are generalized to the context of rings with von Neumann regular total quotient rings. A (commutative unital) ring R is called regular divided if each P ∈ Spec(R)?(Max(R) ∩ Min(R)) is comparable with each principal regular ideal of R. Among rings having von Neumann regular total quotient rings, the regular divided rings are the pullbacks K× _K/P D where K is von Neumann regular, P ∈ Spec(K) and D is a divided domain. Any regular divided ring (for instance, regular comparable ring) with a von Neumann regular total quotient ring is a weak Baer going-down ring. If R is a weak Baer going-down ring and T is an extension ring with a von Neumann regular total quotient ring such that no regular element of R becomes a zero-divisor in T, then R ? T satisfies going-down. If R is a weak Baer ring and P ∈ Spec(R), then R + PR _(P) is a going-down ring if and only if R/P and R _P are going-down rings. The weak Baer going-down rings R such that Spec(R)?Min(R) has a unique maximal element are characterized in terms of the existence of suitable regular divided overrings.     

Straight Rings

A (commutative integral) domain is called a straight domain if A ? B is a prime morphism for each overring B of A; a (commutative unital) ring A is called a straight ring if A/P is a straight domain for all P ∈ Spec(A). A domain is a straight ring if and only if it is a straight domain. The class of straight rings sits properly between the class of locally divided rings and the class of going-down rings. An example is given of a two-dimensional going-down domain that is not a straight domain. The classes of straight rings, of locally divided rings, and of going-down rings coincide within the universe of seminormal weak Baer rings (for instance, seminormal domains). The class of straight rings is stable under formation of homomorphic images, rings of fractions, and direct limits. The “straight domain" property passes between domains having the same prime spectrum. Straight domains are characterized within the universe of conducive domains. If A is a domain with a nonzero ideal I and quotient field K, characterizations are given for A ? (I: _K I) to be a prime morphism. If A is a domain and P ∈ Spec(A) such that A _P is a valuation domain, then the CPI-extension C(P) := A + PA _P is a straight domain if and only if A/P is a straight domain. If A is a going-down domain and P ∈ Spec(A), characterizations are given for A ? C(P) to be a prime morphism. Consequences include divided domain-like behavior of arbitrary straight domains.     

Pseudo-almost valuation domains are quasilocal going-down domains,but not conversely

Abstract  Ayman Badawi has recently introduced the PAVDs, a class of (commutative integral) domains which is found strictly between the class of APVDs (“almost pseudo valuation domains”) and that of the (necessarily quasilocal) domains having a linearly ordered prime spectrum. It is known that the latter class strictly contains the class of quasilocal going-down domains; it is proved that the class of quasilocal going-down domains strictly contains the class of PAVDs. Consequently, each seminormal PAVD is a divided domain. Moreover, for each n, 1 ≤ n ≤ ∞, an example is constructed of a divided domain (necessarily a quasilocal going-down domain) of Krull dimension n which is not a PAVD. Keywords Pseudo-almost valuation domain, Prime ideal, Going-down domain, Divided domain, Quasilocal, Valuation overring, Root extension, Seminormal, D+M construction, Krull dimension Mathematics Subject Classification (2000) Primary 13B24, 13G05, Secondary 13A15, 13F05     

On Finite Maximal Chains of Weak Baer Going-Down Rings

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ? T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T.     

On self-duality of ring extensions over quasi-frobenius rings

ABSTRACT:

Let R be a zero-dimensional SFT-ring. It is proved that the minimal prime ideals of the formal power series ring A=R[[x ₁, …, x_n ]] are the ideals of the form [[x ₁, …, x_n ]], where is a (minimal) prime of R. It follows that A has Krull dimension n and is catenarian. If R?T where T is also a zero-dimensional SFT-ring, the lying-over, going-up, incomparable, and going-down properties are studied for the extension A?T[[x ₁, …, x_n ]].     

Cyclic Maximal Ideals of Rings of Differential Operators over Power Series Rings

For an integral domain R, several necessary and sufficient conditions are given for R to be unibranched inside its absolute integral closure; one such condition is that R^pbe Henselian for each prime ideal P of R. Additional conditions are given in case R is a going-down domain. Unlike the situation in the Noetherian context, such going‐down domains R need not be quasilocal or of Krull dimension at most 1. A number of examples are given for the locally pseudo‐valuation domain case     

A note on complete rings of quotients and McCoy rings

If a (commutative unital) ring $A$ is reduced and coincides with its total quotient ring, then $A$ satisfies Property A (that is, $A$ is a McCoy ring) if and only if the inclusion of $A$ in its complete ring of quotients $C(A)$ is a survival extension. The ??if?? assertion fails if one deletes the hypothesis that $A$ is reduced. This is shown by using the idealization construction to construct a suitable ring $A$ and then identifying its complete ring of quotients (which turns out to be a related idealization). Related characterizations of von Neumann regular rings are also given with the aid of the going-down property GD of ring extensions. For instance, a ring $A$ is von Neumann regular if and only if $A$ is a reduced McCoy ring that coincides with its total quotient ring such that $A \subseteq C(A)$ satisfies GD.