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 the Key Equation Over a Commutative Ring

We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error-locator polynomial is the unique monic minimal polynomial (equivalently, the unique shortest linear recurrence) of the finite sequence of syndromes and that it can be obtained by Algorithm MR of Norton.WhenR is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but that all the minimal polynomials coincide modulo the maximal ideal ofR . We characterise the set of minimal polynomials when R is a Hensel ring. We also apply these results to decoding alternant codes over a local ring R: it is enough to find any monic minimal polynomial over R and to find its roots in the residue field. This gives a decoding algorithm for alternant codes over a finite chain ring, which generalizes and improves a method of Interlando et. al. for BCH and Reed-Solomon codes over a Galois ring.     

Pure Injective Modules over a Commutative Valuation Domain

Using geometrical invariants we classify those pure injective modules over a commutative valuation domain which are envelopes of one element.     

On Finite Saturated Chains of Overrings

If a domain R, with quotient field K, has a finite saturated chain of overrings from R to K, then the integral closure of R is a Prüfer domain. An integrally closed domain R with quotient field K has a finite saturated chain of overrings from R to K with length m ≥ 1 iff R is a Prüfer domain and |Spec(R)| =m + 1. In particular, we prove that a domain R has a finite saturated chain of overrings from R to K with length dim(R) iff R is a valuation domain and that an integrally closed domain R has a finite saturated chain of overrings from R to K with length dim (R) +1 iff R is a Prüfer domain with exactly two maximal ideals such that at most one of them fails to contain every non-maximal prime. The relationship with maximal non-valuation subrings is also established.     

Counting intermediate rings in normal pairs

Let RS be an extension of integral domains. If each intermediate ring in this extension is integrally closed in S, then (R,S) is called a normal pair. We investigate in this work the set of intermediate rings in such ring extensions. We establish several results and equations concerning the cardinality of the set of intermediate rings. In particular, we give a way to compute the number of intermediate rings in normal pairs with only finitely many intermediate rings.     

On the Pure-injectivity Profile of a Ring

An analog of the injective profile of a ring, with relative injectivity replaced by relative pure-injectivity, is investigated. Emphasis is placed on comparing and contrasting the properties of both profiles. In particular, the analog in this context of the notion of poor modules is considered and properties of pure-injectively poor modules are determined. While we do not know of any ring that does not have pure-injectively poor modules, their existence has not been determined in general. Rings having pure-injectively poor modules of various types are characterized.     

Transfer Results for the FIP and FCP Properties of Ring Extensions

For an extension E: R ? S of (commutative) rings and the induced extension F: R(X) ? S(X) of Nagata rings, the transfer of the FCP and FIP properties between E and F is studied. Then F has FCP ? E has FCP. The extensions E for which F has FIP are characterized. While E has FIP whenever F has FIP, the converse fails for certain subintegral extensions; it does hold if E is integrally closed, seminormal, or subintegral with R quasi-local having infinite residue field. If F has FIP, conditions are given for the sets of intermediate rings of E and F to be order-isomorphic.     

Hopf Algebra Dual to a Polynomial Algebra over a Commutative Ring

For a polynomial algebra in several variables over a commutative ring R with a Hopf algebra structure the existence of the dual Hopf algebra is proved.     

The Leading Ideal of a Complete Intersection of Height Two in a 2-Dimensional Regular Local Ring

Let (S,𝔫) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I* be the leading ideal of I in the associated graded ring gr𝔫(S), and set R = S/I and 𝔪 = 𝔫/I. In Goto et al. (2007 Goto , S. , Heinzer , W. , Kim , M.-K. ( 2007 ). The leading ideal of a complete intersection of height two, II . J. Algebra 312 : 709 – 732 . [Google Scholar]), we prove that if μ _G (I*) = n, then I* contains a homogeneous system {ξ _i }_1≤i≤n of generators such that deg ξ _i  + 2 ≤ deg ξ _i+1 for 2 ≤ i ≤ n ? 1, and ht _G (ξ₁, ξ₂,…, ξ _n?1) = 1, and we describe precisely the Hilbert series H(gr𝔪(R), λ) in terms of the degrees c _i of the ξ _i and the integers d _i , where d _i is the degree of D _i  = GCD(ξ₁,…, ξ _i ). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c ₁ + 1, an ascending sequence of positive integers (c ₁, c ₂,…, c _n ), and a descending sequence of integers (d ₁ = c ₁, d ₂,…, d _n  = 0) such that c _i+1 ? c _i  > d _i?1 ? d _i  > 0 for each i with 2 ≤ i ≤ n ? 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.     

THREE NOTES ON NICE EXTENSION DOMAINS

The first note shows that the integral closure L′ of certain localities L over a local domain R are unmixed and analytically unramified, even when it is not assumed that R has these properties. The second note considers a separably generated extension domain B of a regular domain A, and a sufficient condition is given for a prime ideal p in A to be unramified with respect to B (that is, p B is an intersection of prime ideals and B/P is separably generated over A/p for all P ∈ Ass (B/p B)). Then, assuming that p satisfies this condition, a sufficient condition is given in order that all but finitely many q ∈ S = {q ∈ Spec(A), p ? q and height(q/p) = 1} are unramified with respect to B, and a form of the converse is also considered. The third note shows that if R′ is the integral closure of a semi-local domain R, then I(R) = ∩{R′ _p′ ;p′ ∈ Spec(R′) and altitude(R′/p′) = altitude(R′) ? 1} is a quasi-semi-local Krull domain such that: (a) height(N ^*) = altitude(R) for each maximal ideal N ^* in I(R); and, (b) I(R) is an H-domain (that is, altitude(I(R)/p ^*) = altitude(I(R)) ? 1 for all height one p ^* ∈ Spec(I(R))). Also, K = ∩{R _p ; p ∈ Spec(R) and altitude(R/p) = altitude(R) ? 1} is a quasi-semi-local H-domain such that height (N) = altitude(R) for all maximal ideals N in K.