Dualities and the duality density theorems

Dualities \({\langle,\, \rangle:S \times T \rightarrow K}\) for modules S =  _R S, T = T _D and bimodule K =  _R K _D over rings R, D are non-degenerate left dense K-pairings of S, T intertwined per adjoint, classification, and Galois correspondence theorems. Dualities are abundant per density theorems inspired by those of Jacobson and Chevalley. Duality theory generalizes classical duality theory and leads to a theory of duality semi-simplicity of rings R and R-modules S. The finite dimensional duality semi-simple algebras R are classified in terms of semi-simple algebras and bipolar algebras.     

Serre dimension of monoid algebras

Let R be a commutative Noetherian ring of dimension d, M a commutative cancellative torsion-free monoid of rank r and P a finitely generated projective R[M]-module of rank t. Assume M is Φ-simplicial seminormal. If \(M\in \mathcal {C}({\Phi })\), then Serre dim R[M]≤d. If r≤3, then Serre dim R[int(M)]≤d. If \(M\subset \mathbb {Z}_{+}^{2}\) is a normal monoid of rank 2, then Serre dim R[M]≤d. Assume M is c-divisible, d=1 and t≥3. Then P?∧ ^t P⊕R[M] ^t?1. Assume R is a uni-branched affine algebra over an algebraically closed field and d=1. Then P?∧ ^t P⊕R[M] ^t?1.     

Tilting Modules Arising from Ring Epimorphisms

We show that a tilting module T over a ring R admits an exact sequence 0?→?R?→?T ₀?→?T ₁?→?0 such that \(T_0,T_1\in\text{Add}(T)\) and Hom _R (T ₁,T ₀)?=?0 if and only if T has the form S?⊕?S/R for some injective ring epimorphism λ : R?→?S with the property that \(\text{Tor}_1^R(S,S)=0\) and pdS _R ?≤?1. We then study the case where λ is a universal localization in the sense of Schofield (1985). Using results by Crawley-Boevey (Proc Lond Math Soc (3) 62(3):490–508, 1991), we give applications to tame hereditary algebras and hereditary noetherian prime rings. In particular, we show that every tilting module over a Dedekind domain or over a classical maximal order arises from universal localization.     

Posets with Cover Graph of Pathwidth two have Bounded Dimension

Joret, Micek, Milans, Trotter, Walczak, and Wang recently asked if there exists a constant d such that if P is a poset with cover graph of P of pathwidth at most 2, then dim(P)=d. We answer this question in the affirmative by showing that d=17 is sufficient. We also show that if P is a poset containing the standard example S ₅ as a subposet, then the cover graph of P has treewidth at least 3.     

Functional Identities and Rings of Quotients

The fundamental theorem on functional identities states that a prime ring R with \(\deg (R)\ge d\) is a d-free subset of its maximal left ring of quotients Q _{m l} (R). We consider the question whether the same conclusion holds for symmetric rings of quotients. This indeed turns out to be the case for the maximal symmetric ring of quotients Q _{m s} (R), but not for the symmetric Martindale ring of quotients Q _s (R). We show, however, that if the maps from the basic functional identities have their ranges in R, then the maps from their standard solutions have their ranges in Q _s (R). We actually prove a more general theorem which implies both aforementioned results. Its proof is somewhat shorter and more compact than the standard proof used for establishing d-freeness in various situations.     

On the singularity of Quillen metrics

Let π: X → S be a holomorphic map from a compact Kähler manifold (X,g _X ) to a compact Riemann surface S. Let Σ_π be the critical locus of π and let Δ  =  π(Σ_π) be the discriminant locus. Let (ξ, h _ξ) be a holomorphic Hermitian vector bundle on X. We determine the singularity of the Quillen metric on det Rπ_*ξ near Δ with respect to g _X | _TX/S and h _ξ.     

On the minimum size of binary codes with length 2<Emphasis Type="Italic">R</Emphasis> + 4 and covering radius <Emphasis Type="Italic">R</Emphasis>

The minimum size of a binary code with length n and covering radius R is denoted by K(n, R). For arbitrary R, the value of K(n, R) is known when n ≤  2R +  3, and the corresponding optimal codes have been classified up to equivalence. By combining combinatorial and computational methods, several results for the first open case, K(2R +  4, R), are here obtained, including a proof that K(10, 3) =  12 with 11481 inequivalent optimal codes and a proof that if K(2R +  4, R) <  12 for some R then this inequality cannot be established by the existence of a corresponding self-complementary code.     

Regular Apeirotopes of Dimension and Rank 4

In previous papers, all the four-dimensional (finite) regular polytopes have been classified, as well as the regular apeirotopes of full rank (that is, of rank 5). Of the two problems in \(\mathbb{E}^{4}\) thus left open (namely, regular apeirotopes of ranks 3 and 4), this paper describes the regular apeirotopes of rank 4. The methods employed here are somewhat different from those in earlier work; while knowledge of the possible dimension vectors (dim?R ₀,…,dim?R ₃) of the mirrors R ₀,…,R ₃ of the generating reflexions of the symmetry groups plays a rôle, the crystallographic restriction leads to a considerable emphasis being placed on the vertex-figures.     

Higher Algebraic <Emphasis Type="Italic">K</Emphasis>-theory of Ring Epimorphisms

Given a homological ring epimorphism from a ring R to another ring S, we show that if the left R-module S has a finite-type resolution, then the algebraic K-group K _n (R) of R splits as the direct sum of the algebraic K-group K _n (S) of S and the algebraic K-group K _n (R) of a Waldhausen category R determined by the ring epimorphism. This result is then applied to endomorphism rings, matrix subrings, rings with idempotent ideals, and universal localizations which appear often in representation theory and algebraic topology.     

Unbounded disjointness preserving linear functionals and operators

Let E and F be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice E, which shows that in this case the unbounded disjointness preserving operators from \({E\to F}\) separate the points of E. Then we show that every disjointness preserving operator \({T:E\to F}\) is norm bounded on an order dense ideal. In case E has order continuous norm, this implies that every unbounded disjointness preserving map \({T:E\to F}\) has a unique decomposition T = R + S, where R is a bounded disjointness preserving operator and S is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that E = C(X), with X a compact Hausdorff space, we show that every disjointness preserving operator \({T:C(X)\to F}\) is norm bounded on a norm dense sublattice algebra of C(X), which leads then to a decomposition of T into a bounded disjointness preserving operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals.     

Generators for decompositions of tensor products of modules

If K is a field of finite characteristic p, G is a cyclic group of order q = p ^α , U and W are indecomposable KG-modules, and p ≥ dim U + dim W ? 1, we describe how to find a generator for each of the indecomposable components of the KG-module \({U \otimes W}\).     

The regular digraph of ideals of a commutative ring

Let R be a commutative ring and Max?(R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by \(\overrightarrow{\Gamma_{\mathrm{reg}}}(R)\), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γ_reg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian ring R, we prove that |Max?(R)|?1≦ω(Γ_reg(R))≦|Max?(R)| and \(\chi(\Gamma_{\mathrm{ reg}}(R)) = 2|\mathrm{Max}\, (R)| -k-1\), where k is the number of fields, appeared in the decomposition of R to local rings. Among other results, we prove that \(\overrightarrow{\Gamma_{\mathrm{ reg}}}(R)\) is strongly connected if and only if R is an integral domain. Finally, the diameter and the girth of the regular graph of ideals of Artinian rings are determined.     

Generalization of CS condition

Let R be an associative ring with identity. An R-module M is called an NCS module if C (M)∩S(M) = {0}, where C (M) and S(M) denote the set of all closed submodules and the set of all small submodules of M, respectively. It is clear that the NCS condition is a generalization of the well-known CS condition. Properties of the NCS conditions of modules and rings are explored in this article. In the end, it is proved that a ring R is right Σ-CS if and only if R is right perfect and right countably Σ-NCS. Recall that a ring R is called right Σ-CS if every direct sum of copies of R_R is a CS module. And a ring R is called right countably Σ-NCS if every direct sum of countable copies of R_R is an NCS module.     

Power-Commuting Generalized Skew Derivations in Prime Rings

Let R be a non-commutative prime ring of characteristic different from 2 with extended centroid C, F ≠ 0 a generalized skew derivation of R, and n ≥ 1 such that [F(x), x] ⁿ  = 0, for all x ∈ R. Then there exists an element λ ∈ C such that F(x) = λx, for all x ∈ R.     

Maximal invariant subspaces for a class of operators

In this note, we characterize maximal invariant subspaces for a class of operators. Let T be a Fredholm operator and \(1-TT^{*}\in\mathcal{S}_{p}\) for some p≥1. It is shown that if M is an invariant subspace for T such that dim?M ? TM<∞, then every maximal invariant subspace of M is of codimension 1 in M. As an immediate consequence, we obtain that if M is a shift invariant subspace of the Bergman space and dim?M ? zM<∞, then every maximal invariant subspace of M is of codimension 1 in M. We also apply the result to translation operators and their invariant subspaces.     

Eulerian graphs and automorphisms of a maximal graph

Let R be a commutative ring with identity. Let Γ(R) denote the maximal graph corresponding to the non-unit elements of R, i.e., Γ(R) is a graph with vertices the non-unit elements of R, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of R containing both. In this paper, we have shown that, for any finite ring R which is not a field, Γ(R) is a Euler graph if and only if R has odd cardinality. Moreover, for any finite ring R ? R ₁×R ₂× · · · ×R _n, where the R _i is a local ring of cardinality p _i ^αi for all i, and the p _i’s are distinct primes, it is shown that Aut(Γ(R)) is isomorphic to a finite direct product of symmetric groups. We have also proved that clique(G(R)’) = χ(G(R)’) for any semi-local ring R, where G(R)’ denote the comaximal graph associated to R.     

Finitistic dimension and restricted injective dimension

We study the relations between finitistic dimensions and restricted injective dimensions. Let R be a ring and T a left R-module with A = End _R T. If _R T is selforthogonal, then we show that rid(T _A ) ? findim(A _A ) ? findim( _R T) + rid(T _A ). Moreover, if R is a left noetherian ring and T is a finitely generated left R-module with finite injective dimension, then rid(T _A ) ? findim(A _A ) ? fin.inj.dim( _R R)+rid(T _A ). Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.     

On maximal subgroups of the group of recursive permutations

The group G _R of all permutations belonging to R is considered for any partially recursively closed class of functions R. It is proved that the group G _R has a continuum number of maximal subgroups. Examples of constructive maximal subgroups of G _R are given.     

On the Dunford Property (<Emphasis Type="Italic">C</Emphasis>) for Bounded Linear Operators <Emphasis Type="Italic">RS</Emphasis> and <Emphasis Type="Italic">SR</Emphasis>

In this paper we show that if \({S\in L(X,Y)}\) and \({R\in L(Y,X),}\) X and Y complex Banach spaces, then the products RS and SR share the Dunford property (C). We also study property (C) for R, S, RS and \({SR \in L(X)}\) in the case that R and S satisfies the operator equations RSR = R ² and SRS = S ².