GENERALIZED MATLIS DUALITY AND LINEAR COMPACTNESS

Uniquely separable extensions

Torsion theories over commutative rings

This paper centers around the theorem that a commutative ring R R is noetherian if every R P , P {R_P},P prime, is noetherian and every finitely generated ideal of R R has only finitely many weak-Bourbaki associated primes. A more precise local version of this theorem is also given, and examples are presented to show that the assumption on the weak-Bourbaki primes cannot be deleted nor replaced by the assumption that Spec ( R ) (R) is noetherian. Moreover, an alternative statement of the theorem using a variation of the weak-Bourbaki associated primes is investigated. The proof of the theorem involves a knowledge of the behavior of associated primes of an ideal under quotient ring extension, and the paper concludes with some remarks on this behavior in the more general setting of flat ring extensions.

The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the $S$-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the $S$-variant of Eakin–Nagata–Eisenbud theorem for general rings by using $S$-Noetherian modules. We also show that every right $S$-Noetherian domain is right Ore, which is embedded into a division ring. For a right $S$-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right $S$-Noetherian or right Noetherian. As applications, the $S$-variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.

Let R be a commutative ring with identity. A short exact sequence of R-modules is said to be neat (respectively, -pure) if the sequence (respectively, ) is exact for every simple R-module S. The characterization of the integral domains over which neatness and -purity coincide has been given by László Fuchs: they are the integral domains where every maximal ideal is projective (and so also finitely generated). We show that for a commutative ring R, every maximal ideal of R is finitely generated and projective if and only if R has projective socle and neatness and -purity coincide. For a commutative ring R, we prove that neatness and -purity coincide if and only if every maximal ideal of R is finitely generated and the unique maximal ideal PP of the local ring RP is a principal ideal for every maximal ideal P of R. This result is proved firstly over commutative local rings and then using localization over any commutative ring. The Auslander-Bridger transpose of simple modules is used in proving these equivalences because it is an essential tool for the passage between proper classes of short exact sequences of modules that are projectively generated and these that are flatly generated by a set of finitely presented modules. For a commutative ring R, we prove that neatness and -purity coincide if and only if every simple R-module S is finitely presented and an Auslander-Bridger transpose of S is projectively equivalent to S.

Let S be a cancellative monoid with quotient group of torsion-free rank a.We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a.Assume that R is a commutative unitary ring and G is an abelian group.The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the group ring R[G] should be a Hilbert ring.Matsuda has considered this question in [Ml, 6].His results show that for G finitely generated, R[G] and R are simultaneously Hilbert rings; if G is not finitely generated, then R[G] a Hilbert ring implies that R is a Hilbert ring, but the converse fails.In [M2, 5] Matsuda considers briefly the corresponding Hilbert-ring-characterization problem for a monoid ring R[S], where S is torsion-free and cancellative.Let a be the torsion-free rank of G and let {Xj)lsl be a set of indeterminates over R of cardinality a.In Corollary 1 we show that R[G] and P[{^,},e/] are simultaneously Hilbert rings.While the problem of determining equivalent conditions for P[{Ar,}/e/] to be a Hilbert ring has not been completely resolved, it has been worked on extensively [Kr, Go, L, Gl, H], and a significant body of positive results exists concerning this problem.The proof of Theorem 1 suggests the following conjecture: if S is a cancellative commutative monoid with quotient group G, then R[S] is a Hilbert ring if and only if R[G] is a Hilbert ring; this result is established in Theorem 2.All monoids considered are assumed to be commutative, and rings are assumed to be commutative and unitary.The statement of Theorem 1 uses the following terminology.For a cardinal number a, an extension ring T of R is said to be a-generated over R if T is generated over R by a set of cardinality at most a.Theorem 1. Assume that X = {XA is a set of indeterminates of cardinality a over the ring R. Denote by Za the direct sum of a copies of the additive group Z of integers.The following conditions are equivalent.(1) R[X] is not a Hilbert ring.(2) There exists a prime ideal P of R such that R/P admits an a-generated extension ring that is a G-domain, but not afield.(3) The group ring R[Za] is not a Hilbert ring._

INTRODUCTION In the broadest sense, this is a study of commutative rings which satisfy the (finitely) pseudo-Froben[us (or (F)PF) condition: All (finitely generated) faithful modules generate the category mod-R of all R-modules. These rings include: Pr[[fer rings, almost maximal valuation rings, self-injective rings, e.g. , quasi-Frobenius (QF) and pseudo-Frobenius (PF) rings, and finite products of these. (In fact, any product of commutative FPF rings is FPF [34]; hence, any product of commutative PF rings is FPF (cf. §9).) If R is FPF, so is its (classical)ring of quotients Qc~(R) and its maximal quotient ring Qmax(R). All known FPF rings are (classically) quotientinjective in the sense that Qcf is injective. 2 We conjecture that all FIDF rings are quot[ent-injective, and prove this in the three cases: (i) local rings (Proposition 7 and Theorem 9B): (2) Noether[an rings (Theorem ii; Endo' s Theorem [25J~ (3) reduced rings (Proposition 3B and Theorem 4). Moreover, any FPF commutative ring R splits, R = R 1 X R2, where R 1 is semihered[tary, and R 2 has essential nilradical. (If R is semilocal or Noetherian, then R 2 is injective. ) Thus any reduced FPF ring has regular [njective Qc~' and conversely any quotient-injective sem[hered[tary ring is FPF (Theorem 4). A ring iS pre-(l~PF iff all(finitely generated)faithful ideals are generators, and we

Let [Formula: see text] be an odd prime number and [Formula: see text]. Assume that either [Formula: see text]i[Formula: see text] [Formula: see text] or [Formula: see text]ii[Formula: see text] [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text] [Formula: see text]respectively, assume that [Formula: see text] and [Formula: see text] is congruent to either [Formula: see text] or [Formula: see text] modulo [Formula: see text]. Then there exist exactly five [Formula: see text]respectively, exactly seven[Formula: see text] isomorphism classes of rings [Formula: see text] for which there exists a tower [Formula: see text] of ramified [Formula: see text]integral minimal[Formula: see text] ring extensions such that [Formula: see text] is the only ring properly contained between [Formula: see text] and [Formula: see text]. We produce a set of isomorphism class representatives of such [Formula: see text] and, for each such [Formula: see text], we show that, up to isomorphism, the corresponding ring [Formula: see text] is the idealization [Formula: see text]. One consequence, for each integer [Formula: see text] whose prime-power factorization [Formula: see text] (with pairwise distinct prime numbers [Formula: see text] and positive integers [Formula: see text]) satisfies [Formula: see text] for all [Formula: see text], is a classification, up to isomorphism, of the rings that have cardinality [Formula: see text] and exactly two proper (unital) subrings.

The property of a generating set of a polynomial ideal or of an ideal of a free associative algebra over a commutative ring to be its Gröbner basis is kept by a flat (but not any) extension of the ground ring. The converse is proved in the broader context of standard bases for filtered modules.

Descent theory and Amitsur cohomology of triples

Let [Formula: see text] be a (unital) extension of (commutative) rings. We say that [Formula: see text] is a maximal non-quasi-local (respectively, non-PVD) subring of [Formula: see text] if [Formula: see text] is not quasi-local (respectively, PVD) and each subring of [Formula: see text] properly containing [Formula: see text] is quasi-local (respectively, PVD). The aim of this paper is to study this kind of ring extensions and to investigate the structure of the intermediate rings between [Formula: see text] and [Formula: see text].

In this paper, we introduce a new class of ring called regular S-Noetherian ring, which is a weak version of S-Noetherian ring property. Any S-Noetherian ring is naturally a regular S-Noetherian ring, and in the domain context, these two forms coincide. We study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamated duplication of a ring along an ideal. Our results generate new families of examples of non-S-Noetherian regular S-Noetherian rings.

For an extension A ⊆ B of commutative rings, we present a sufficient condition for the ring [[A S, ≤ ]] of generalized power series to be weakly normal (resp., strongly t-closed) in [[B S, ≤ ]], where (S, ≤) be a torsion-free cancellative strictly ordered monoid.As a corollary, it can be applied to the ring of power series in infinitely many indeterminates as well as in finite indeterminates.

For arbitrary categories and and an arbitrary functor the author introduces the notion of an -normal object and proves a general type of fundamental theorem of Galois theory for such objects. It is shown that the normal extensions of commutative rings and central extensions of multi-operator groups are special cases of -normal objects.Bibliography: 14 titles.

We consider extensions of unital commutative rings. We define an extension R ↪ S R\hookrightarrow S to be a p p -extension if every principally generated ideal of S S is generated by an element of R R . Examples are plentiful and localizations of regular multiplicative sets are p p -extensions. We develop the theory of p p -extensions.