Chain and distributive coalgebras

We show that coalgebras whose lattice of right coideals is distributive are coproducts of coalgebras whose lattice of right coideals is a chain. Those chain coalgebras are characterized as finite duals of Noetherian chain rings whose residue field is a finite dimensional division algebra over the base field. They also turn out to be coreflexive. Infinite dimensional chain coalgebras are finite duals of left Noetherian chain domains. Given any finite dimensional division algebra D and D-bimodule structure on D, we construct a chain coalgebra as a cotensor coalgebra. Moreover if D is separable over the base field, every chain coalgebra of type D can be embedded in such a cotensor coalgebra. As a consequence, cotensor coalgebras arising in this way are the only infinite dimensional chain coalgebras over perfect fields. Finite duals of power series rings with coeficients in a finite dimensional division algebra D are further examples of chain coalgebras, which also can be seen as tensor products of D^∗, and the divided power coalgebra and can be realized as the generalized path coalgebra of a loop. If D is central, any chain coalgebra is a subcoalgebra of the finite dual of D[[x]].     

On radicals of module coalgebras

We introduce the notion of an idempotent radical class of module coalgebras over a bialgebra B. We prove that if R is an idempotent radical class of B-module coalgebras, then every B-module coalgebra contains a unique maximal B-submodule coalgebra in R. Moreover, a B-module coalgebra C is a member of R if, and only if, DB is in R for every simple subcoalgebra D of C. The collection of B-cocleft coalgebras and the collection of H-projective module coalgebras over a Hopf algebra H are idempotent radical classes. As applications, we use these idempotent radical classes to give another proofs for a projectivity theorem and a normal basis theorem of Schneider without assuming a bijective antipode.     

Examples of Generic Lattice Ideals of Codimension 3

A commutative ring R is said to be strongly Hopfian if the chain of annihilators ann(a) ? ann(a ²) ? … stabilizes for each a ∈ R. In this article, we are interested in the class of strongly Hopfian rings and the transfer of this property from a commutative ring R to the ring of the power series R[[X]]. We provide an example of a strongly Hopfian ring R such that R[[X]] is not strongly Hopfian. We give some necessary and sufficient conditions for R[[X]] to be strongly Hopfian.     

The strong Brauer group of a cocommutative coalgebra

Using strong equivalences for coalgebras we define the strong Brauer group of a cocommutative coalgebra C, which is a subgroup of the Brauer group of C. In general there is not a good relation between the Brauer group of a coalgebra and the Brauer group of the dual algebra C∗, the former is not even a torsion group. We find that this subgroups embeds in the Brauer group of C∗. A key tool in this result is the use of techniques from torsion theory. Some cases where both subgroups coincide are shown, for example, C being coreflexive.     

Covering Coalgebras and Dual Non-singularity

Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Colocalisation in comodule categories has been investigated by some authors (see Jara et al., Commun. Algebra, 34(8):2843–2856, 2006 and Nastasescu and Torrecillas, J. Algebra, 185:203–220, 1994). Here we look at possible coalgebra covers π : D → C that could play the rôle of a coalgebra colocalisation. Codense covers will dualise dense (or rational) extensions; a maximal codense cover construction for coalgebras with projective covers is proposed. We also look at a dual non-singularity concept for modules which turns out to be the comodule-theoretic property that turns the dual algebra of a coalgebra into a non-singular ring. As a corollary we deduce that hereditary coalgebras and hence path coalgebras are non-singular in the above sense. We also look at coprime coalgebras and Hopf algebras which are non-singular as coalgebras.     

Strongly rational comodules and semiperfect Hopf algebras over QF rings

Let C be a coalgebra over a QF ring R. A left C-comodule is called strongly rational if its injective hull embeds in the dual of a right C-comodule. Using this notion a number of characterizations of right semiperfect coalgebras over QF rings are given, e.g., C is right semiperfect if and only if C is strongly rational as left C-comodule. Applying these results we show that a Hopf algebra H over a QF ring R is right semiperfect if and only if it is left semiperfect or — equivalently — the (left) integrals form a free R-module of rank 1.     

Isonoetherian power series rings

Facchini and Nazemian proved that a valuation domain is isonoetherian if and only if it is discrete of Krull dimension ≤2 and they showed that this cannot be generalized from the local case to the global case: the 2-dimensional generalized Dedekind domain ?+X?[[X]] is not isonoetherian. Let D be an integral domain with quotient field K. We provide necessary and sufficient conditions on D and K, so that the ring D+XK[[X]] is isonoetherian. We deduce that if D is integrally closed, then D+XK[[X]] is isonoetherian if and only if D is a semi-local principal ideal domain.     

The Splitting Problem for Coalgebras: A Direct Approach

In this note we give a new and elementary proof of a result of Năstăsescu and Torrecillas (J. Algebra, 281:144–149, 2004) stating that a coalgebra C is finite dimensional if and only if the rational part of any right module M over the dual algebra is a direct summand in M (the splitting problem for coalgebras). Research supported by a CNCSIS BD-type grant, and by the bilateral project BWS04/04 “New Techniques in Hopf Algebra Theory and Graded Ring Theory” of the Flemish and Romanian governments.     

Structure theorem for dual quasi-hopf bicomodules and its application

We present a structure theorem for dual quasi-Hopf bicomodules, and also obtain the structure theorem C ≌ D ? H for dual quasi-Hopf module coalgebras, where H is a dual quasi-Hopf algebra, C a right H-module coalgebra, and D a left H-comodule coalgebra in the tensor category ^H M induced from C, and D ? H the smash coproduct introduced by Bulacu and Nauwelaerts.     

On power series rings over valuation domains

Let V be a valuation domain and V[[X]] be the power series ring over V. In this paper, we show that if V[[X]] is a locally finite intersection of valuation domains, then V is an SFT domain and hence a discrete valuation domain. As a consequence, it is shown that the power series ring V[[X]] is a Krull domain if and only if V[[X]] is a generalized Krull domain if and only if V[[X]] is an integral domain of Krull type (or equivalently, a PvMD of finite t-character) if and only if V is a discrete valuation domain with Krull dimension at most one.     

Power series rings and projectivity

We show that a formal power series ring A[[X]] over a noetherian ring A is not a projective module unless A is artinian. However, if (A,) is any local ring, then A[[X]] behaves like a projective module in the sense that Ext^p_A(A[[X]], M)=0 for all -adically complete A-modules. The latter result is shown more generally for any flat A-module B instead of A[[X]]. We apply the results to the (analytic) Hochschild cohomology over complete noetherian rings. The authors were partly supported by NSERC grant 3-642-114-80 and by the DFG Schwerpunkt ``Global Methods in Complex Geometry'.     

When does the rational submodule split off?

A coalgebraC is said to have the splitting property if the maximal rational submoduleRat(M) of any leftC ^*-moduleM is a direct, summand of it. In this paper we prove that a coalgebraC satisfying this property is finite dimensional. Cocommutative coalgebras such thatRat(M) is a direct summand for any finitely generated leftC ^*-moduleM are explicitly described.

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Sunto Si dice che una coalgebraC ha la proprietà, di spezzarsi se il sottomodulo razionale massimaleRat(M) di ogniC ^*-modulo sinistroM è un suo addendo diretto. In questo articolo dimostriamo che una coalgebraC che soddisfa questa propretà è di dimensione finita. Si descrivono esplicitamente le coalgebre cocommutative tali cheRat(M) è un addendo diretto per ogniC ^*-modulo sinistro finitamente generato.

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Supported by grant BFM2002-02717 from DGES.     

Describing units of integral group rings of some 2-groups

Let R be a ring and M a right R-module. The module M is a CS-module or satifies (C₁) if every submodule is essential in a direct summand of M. In this note we investigate two generalizations of CS-modules.     

The structure theorem for weak module coalgebras

Let H be a weak Hopf algebra, let C be a weak right H-module coalgebra, and let $ \bar C = {C \mathord{\left/ {\vphantom {C C}} \right. \kern-0em} C} \cdot Ker \sqcap ^L $ . We prove a structure theorem for weak module coalgebras, namely, C is isomorphic as a weak right H-module coalgebra to a weak smash coproduct $ \bar C $ × H defined on a k-space $$ \{ \Sigma c_{(0)} \otimes h_2 \varepsilon (c_{( - 1)} h_1 )|c \in C,h \in H\} $$ if there exists a weak right H-module coalgebra map ?: C → H.     

t-closed rings of formal power series

Let A ì BA\subset B be rings. We say that A is t-closed in B, if for each a ? Aa\in A and b ? Bb\in B such that b²-ab,b³-ab² ? Ab^2-ab,b^3-ab^2\in A, then b ? Ab\in A. We present a sufficient condition for the ring A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] to be t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]]. By an example, we show that our condition is not necessary. Even though the question is still open, some important cases are treated. For example, if A ì BA\subset B is an integral extension, or if A is p-injective, then A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] is t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]] if and only if A is t-closed in B.     

The catenarian property of power series rings over a globalized pseudo-valuation domain

LetR be a locally finite dimensional globalized pseudo-valuation domain with Arnold’s SFT-property. It is shown that the power series ringR[[X]] is catenarian. Examples of non catenarianR[[X]] (resp.R[X]) withR[X] (resp.R[[X]]) catenarian are also given. Supported by 60% MPI Research Fund.     

Representation-Directed Incidence Coalgebras of Intervally Finite Posets and the Tame-Wild Dichotomy

Incidence coalgebras C = K ^□ I of intervally finite posets I that are representation-directed are characterized in the article, and the posets I with this property are described. In particular, it is shown that the coalgebra C = K ^□ I is representation-directed if and only if the Euler quadratic form q _C : ?^(I) → ? of C is weakly positive. Every such a coalgebra C is tame of discrete comodule type and gl. dimC ≤ 2. As a consequence, we get a characterization of the incidence coalgebras C = K ^□ I that are left pure semisimple in the sense that every left C-comodule is a direct sum of finite dimensional subcomodules. It is shown that every such coalgebra C = K ^□ I is representation-directed and gl. dimC ≤ 2. Finally, the tame-wild dichotomy theorem is proved, for the coalgebras K ^□ I that are right semiperfect.     

Special isomorphisms of F[x 1,..., x n ] preserving GCD and their use

On the ring R = F[x ₁,..., x _n ] of polynomials in n variables over a field F special isomorphisms A’s of R into R are defined which preserve the greatest common divisor of two polynomials. The ring R is extended to the ring S: = F[[x ₁,..., x _n ]]⁺ and the ring T: = F[[x ₁,..., x _n ]] of generalized polynomials in such a way that the exponents of the variables are non-negative rational numbers and rational numbers, respectively. The isomorphisms A’s are extended to automorphisms B’s of the ring S. Using the property that the isomorphisms A’s preserve GCD it is shown that any pair of generalized polynomials from S has the greatest common divisor and the automorphisms B’s preserve GCD. On the basis of this Theorem it is proved that any pair of generalized polynomials from the ring T = F[[x ₁,..., x _n ]] has a greatest common divisor.