Lifting trees of prime ideals to bezout extension domains

We prove that if an extension R ? T of commutative rings satisfies the going-up property, then any tree of prime ideals of R with at most two branches or in which each branch has finite length is covered by some corresponding tree of prime ideals of T. In particular, if R ? T is an integral extension and R is Noetherian, then each tree in Spec(R) can be covered by a tree in Spec(T). We also prove that if R is an integral domain, then each tree T in Spec(/2) can be covered by a tree in Spec(T) for some Bezout domain T containing R. If T has only finitely many branches, it can further be arranged that the Bezout domain T be an overring of R. However, in general, it cannot be arranged that T be covered from a Prüfer overring of R, thus answering negatively a question of D D. Anderson.     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

Domains whose overrings satisfy accp

An integral domain D satisfies ACC on principal ideals (ACJCP) if there does not exist an infinite strictly ascending chain of principal ideals of D. Any Noetherian domain, in particular any Dedekind domain, satisfies ACCP. In this note we prove the following theorem: Let D be an integral domain. Then the integral closure of D is a Dedekind domain if and only if every overring of D (ring between D and its quotient field) satisfies ACCP.     

On Local ☆-Completely Integrally Closed Domains

Let ☆ be a star operation on an integral domain R. The domain R is a ☆-CICD if (AA ^?1)^☆ = R for all nonzero (fractional) ideals A of R. In this article, we prove that, if the maximal ideal of a local ☆-CICD is a ☆-ideal, then R is ☆-principal ideal domain. We also establish that any ☆-CICD R is locally a PID when ☆ is induced by the localizations at prime ideals of R.     

On a Field-Theoretic Invariant for Extensions of Commutative Rings

Abstract

If R ? T is an extension of (commutative integral) domains, Λ(T/R) is defined as the supremum of the lengths of chains of intermediate fields in the extension k _R (Q ∩ R) ? k _T (Q), where Q runs over the prime ideals of T. The invariant Λ(T/R) is determined in case R and T are adjacent rings and in case Spec(R) = Spec(T) as sets. It is proved that if R is a domain with integral closure R′, then Λ(T/R) = 0 for all overrings T of R if and only if R′ is a Prüfer domain such that Λ(R′/R) = 0. If R ? T are domains such that the canonical map Spec(T) → Spec(R) is a homeomorphism (in the Zariski topology), then Λ(T/R) is bounded above by the supremum of the lengths of chains of rings intermediate between R and T. Examples are given to illustrate the sharpness of the results.     

D-strong and almostD-strong near-rings

In this paper,D-strong and almostD-strong near-rings have been defined. It has been proved that ifR is aD-strongS-near ring, then prime ideals, strictly prime ideals and completely prime ideals coincide. Also ifR is aD-strong near-ring with identity, then every maximal right ideal becomes a maximal ideal and moreover every 2-primitive near-ring becomes a near-field. Several properties, chain conditions and structure theorems have also been discussed.Most of the parts of this paper are included in author's doctoral dissertation at Sukhadia University Udaipur (1983). The author expresses his gratitude to Dr.S. C. Choudhary for his kind guidance.     

Differential rings and ore extensions: Brown-McCoy rings

We consider here a ringK, a derivationD ofK and the differential polynomial ringR=K[X;D]. The ringK is said to be a Brown-McCoy ring if the prime radical coincides with the Brown-McCoy radical in every homomorphic image ofK. AD-Brown-McCoy ring is defined in a similar way. We prove the following conditions are equivalent: (i)K is aD-Brown-McCoy ring; (ii)R is a Brown-McCoy ring and for every maximal idealM ofR,K/(MνK) is aD-simple ring with 1. In addition, we give some applications and examples on the study of the transfer of the property of being a Brown-McCoy ring betweenK andR. Further, we study the relation between the prime and theD-prime ideals of a differential intermediate extension of a liberal extension. This paper was supported by a fellowship awarded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.     

Chain conditions in modules with krull dimension

Any ring with Krull dimension satisfies the ascending chain condition on semiprime ideals. This result does not hold more generally for modules. In particular if Ris the first Weyl algebra over a field of characteristic 0 then there are Artinian R-modules which do not satisfy the ascending chain condition on prime submodules. However, if Ris a ring which satisfies a polynomial identity then any R-module with Krull dimension satisfies the ascending chain condition on prime submodules, and, if Ris left Noethe-rian, also the ascending chain condition on semiprime submodules.     

Spectra of smash products

LetT=R #H be a smash product whereH is a finite dimensional Hopf algebra. We show that ideals ofT invariant under the dualH* ofH are extended fromH-invariant ideals ofR. This allows us to transport the study of ideals inT to invariant ideals. When the Hopf algebra is pointed the relationship between an ideal and its invariant ideal is shown to be manageable. Restricting to prime ideals, this yields results on the prime spectra ofR andT. We obtain Krull relations forR ⊆T for someH, including Incomparability wheneverH is commutative (or more generally whenH* is pointed after base extension). The results generalize and unify a number of results known in the context of group and restricted Lie actions.     

Pullbacks of Arithmetical Rings

We give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (i.e., a ring whose ideals are totally ordered by inclusion). We also give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (i.e., a ring which is locally a chain ring at every maximal ideal). For any integral domain D with field of fractions K, we characterize all Prüfer domains R between D[X] and K[X] such that the conductor C of K[X] into R is nonzero. As an application, we show that for n ≥ 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.     

Integrally Closed Domains with Only Finitely Many Star Operations

We prove that an integrally closed domain R admits only finitely many star operations if and only if R satisfies each of the following conditions: (1) R is a Prüfer domain with finite character, (2) all but finitely many maximal ideals of R are divisorial, (3) only finitely many maximal ideals of R contain a nonzero prime ideal that is contained in some other maximal ideal of R, and (4) if P ≠ (0) is the largest prime ideal contained in a (necessarily finite) collection of maximal ideals of R, then the prime spectrum of R/P is finite.     

ASSOCIATIONS AND VALUATIONS

Abstract

Right cones are semigroups for which the lattice of right ideals is a chain and a left cancellation law holds; valuation rings, the cones of ordered groups, and initial segments of ordinal numbers are examples. Two such cones are associated if they have isoniorphic lattices of right ideals so that ideals, prime ideals, and completely prime ideals correspond to each other. A list of problems is discussed. In Proposition 3.11 it is proved that the canonical mapping from a right invariant right chain domain R onto the associated right holoid can be extended to a valuation from the skew field Q(R) of quotients of R onto an ordered group if and only if Ja ? aJ for all a ∈ R and J = J(R), the Jacobson radical of R.     

Krull Dimension in Power Series Ring Over an Almost Pseudo-Valuation Domain

Let R be an integral domain. We say that R is a star-domain if R has at least a height one prime ideal and if for each height one prime ideal P of R, R satisfies the acc on P-principal ideals (i.e., ideals of the form aP, a ∈ R). We prove that if R is an APVD with nonzero finite Krull dimension, then the power series ring R[[X]] has finite Krull dimension if and only if R is a residually star-domain (i.e., for each nonmaximal prime ideal P of R, R/P is a star-domain) if and only if R[[X]] is catenarian.     

Über die Distributivität des Idealverbandes eines kommutativen Ringes

LetR be a commutative ring and (R)the lattice of all ideals ofR.. In the first part of this paper we give a sufficient condition for an ideal ofR to belong toD (R) using a certain prime ideal systemP (R). In the second part we investigateD (S) whenS is an integral extension of the integral domainR. An idealI ofS belongs toD (S) if noPP (R) containsIR.     

Four notes on GB-rings

The following results are proved: (1) there exist integrally closed Noetherian domains which are not GB-rings, so the GB-conjecture does not hold; (2) Henselian domains of altitude two and integrally closed Henselian domains of altitude three are GB-rings; and there exist Henselian local domains of altitude three and integrally closed Henselian local domains of altitude four which are not GB-rings; (3) the GB-condition does not descend even for Henselian local domains; and, (4) if (R, M) is a local domain, then D=R[X]_(M,X) is a GB-ring if (and only if) adjacent prime ideals in local integral extension domains of D contract in D to adjacent prime ideals.     

On the Ideals of a Lie Ring of Derivations

Let R be a commutative ring, and D a Lie subring and an R-submodule of Der(R) such that R is D-semiprime (or D-prime). We investigate the structure of the ideals of D as Lie rings. As a consequence, we give a necessary and sufficient condition for the ideals of D to be semiprime (or prime, respectively) Lie rings.     

Notes on a new chain condition for prime ideals

A chain condition intermediate to the catenary property and the chain condition for prime ideals (c.c.) is studied. Like the c.c., the condition is inherited from a semi-local domain R by integral extension domains, by local quotient domains, and by factor domains, and a semi-local ring that satisfies the condition is catenary. (Unlike the c.c., none of these statements is true when R is not semi-local.) A number of characterizations of a semi-local domain that satisfies the condition are given in terms of: integral (respectively, algebraic, transcendental) extension domains, Henselizations, completions, Rees rings, associated graded rings and certain discrete valuation over-rings. Then four of the catenary chain conjectures are characterized in terms of this condition.     

Local bijective gabriel correspondence and relative fully boundedness in τ-noetherian rings

Let Rbe a right τ-noetherian ring, where τ denotes a hereditary torsion theory on the category of right R-modules. It is shown that every essential τ-closed right ideal of every prime homomorphic image of Rcontains a nonzero two-sided ideal if and only if any two τ-torsionfree injective indecomposable right R-modules with identical associated prime ideals are isomorphic, and for any τ-closed prime ideal Pthe annhilator of a finitely generated P-tame right R-module cannot be a prime ideal properly contained in P. Furthermore, if in the last condition finitely generated is replaced by r-noetherian, then all τ-noetherian τ-torsionfree modules turn out to be finitely annihilated.     

Radically perfect prime ideals in polynomial rings

We call an ideal I of a commutative ring R radically perfect if among the ideals of R whose radical is equal to the radical of I the one with the least number of generators has this number of generators equal to the height of I. Let R be a Noetherian integral domain of Krull dimension one containing a field of characteristic zero. Then each prime ideal of the polynomial ring R[X] is radically perfect if and only if R is a Dedekind domain with torsion ideal class group. We also show that over a finite dimensional Bézout domain R, the polynomial ring R[X] has the property that each prime ideal of it is radically perfect if and only if R is of dimension one and each prime ideal of R is the radical of a principal ideal.