Completely integrally closed modules and rings

A ring A is a completely integrally closed right A-module if and only if the maximal right ring of quotients Q _max(A) of A is an injective right A-module and A is a right completely integrally closed subring in Q _max(A). A right Noetherian, right integrally closed ring A is a completely integrally closed right A-module.     

On Chow Groups of G-Graded Rings

Abstract

Let A be a Noetherian ring graded by a finitely generated Abelian group G. It is shown that a Chow group A _?(A) of A is determined by cycles and a rational equivalence with respect to certain G-graded ideals of A. In particular, A _?(A) is isomorphic to the equivariant Chow group of A if G is torsion free.     

Functions realizing as abelian group automorphisms

Let A be a set and f:A→A a bijective function. Necessary and sufficient conditions on f are determined which makes it possible to endow A with a binary operation ? such that (A,?) is a cyclic group and f∈Aut(A). This result is extended to all abelian groups in case |A| = p², p a prime. Finally, in case A is countably infinite, those f for which it is possible to turn A into a group (A,?) isomorphic to ?ⁿ for some n≥1, and with f∈Aut(A), are completely characterized.     

Compact DG modules and Gorenstein DG algebras

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras. When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness. For a finite connected DG algebra A, we prove that D^c(A) and D^c(A ^op) admit Auslander-Reiten triangles if and only if A and A ^op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, D^c(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D_lf^b(A) and D_lf^b (A ^op) instead, when A is a regular DG algebra. This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)     

Generalized Smash Products

In this paper．we study the ring #(D．B)and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all(B．D)-Hopf modules BM^D．Cai and Chen have proved this result in the case B=D=A．Secondly they have proved that if A has a nonzero left integral then A#A^*rat is a dense subring of Endk(A)．We prove that #(A,A) is a dense subring of Endk(Q),where Q is a certain subspace of #(A．A)under the condition that the antipode is bijective(see Theorem18).This condition is weaker than the condition that A has a nonzero integral.It is well known the antipode is bijective in case A has a nonzero integral．Furthermore if A has nonzero left integral,Q can be chosen to be A(see Corollary 19)and #(A,A)is both left and right primitive.Thus A#A^*rat #(A,A)-Endk(A)．Moreover we prove that the left singular ideal of the ring #(A,A)is zero．A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional,namely the ring #(A,A)has a finite uniform dimension．     

Smash Products of H-Simple Module Algebras

Let H be a finite-dimensional Hopf algebra and A a finite-dimensional H-simple left H-module algebra. We show that the smash product A#H is isomorphic to End _A′(V ? H*), where V ≠ 0 is a finite-dimensional left A-module and (A′, V′) the stabilizer of (A, V). As an application it is proved that A#H is isomorphic to a full matrix algebra over A′ when H is semisimple and dim V|dim A.     

Rings over which all modules are <Emphasis Type="Italic">I</Emphasis><Subscript>0</Subscript>-modules

Let A be a ring that does not contain an infinite set of idempotents that are orthogonal modulo the ideal SI(A _A ). It is proved that all A-modules are I ₀-modules if and only if either A is a right semi-Artinian, right V-ring or A/SI(A _A ) is an Artinian serial ring and the square of the Jacobson radical of A/SI(A _A ) isequal to zero. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 5, pp. 193–200, 2007.     

On semiperfect associative pairs

It is known [8] that a semiperfect ring is characterized by the existence of a frame, i.e, a complete set of local orthogonal idempotents. We prove in this paper that a similar behaviour occurs when dealing with an associative pair A, namelyAis semiperfect if and only if Acontains a frame and ā=A/Rad Ais unital. Moreover, we show that, when ā is unital, the existence of a frame for Ais equivalent to the condition that every irreducible right A-module is isomorphic to _e A/e(RadA) for some idempotent e of A.     

Quotients of Incidence Algebras and the Euler Characteristic

In this note, we show that, if A ? kQ _A /I _A is a schurian strongly simply connected algebra given by its normed presentation, and Σ is the unique poset whose Hasse quiver coincides with Q _A , then A ? kΣ if and only if I _A has a generating set consisting of exactly χ(Q _A ) elements, where χ(Q _A ) is the Euler characteristic of Q _A . We also prove that a quotient of an incidence algebra A = kΣ/J is strongly simply connected if and only if A is simply connected and kΣ is strongly simply connected.     

A Note on Relative Efficiency of Axiom Systems

We introduce a notion of relative efficiency for axiom systems. Given an axiom system A_β for a theory T consistent with S¹₂, we show that the problem of deciding whether an axiom system A_α for the same theory is more efficient than A_β is II₂-hard. Several possibilities of speed-up of proofs are examined in relation to pairs of axiom systems A_α, A_β, with A_α ? A_β, both in the case of A_α, A_β having the same language, and in the case of the language of A_α extending that of A_β: in the latter case, letting Pr_α, Pr_β denote the theories axiomatized by A_α, A_β, respectively, and assuming Pr_α to be a conservative extension of Pr_β, we show that if A_α — A_β contains no nonlogical axioms, then A_α can only be a linear speed-up of A_β; otherwise, given any recursive function g and any A_β, there exists a finite extension A_α of A_β such that A_α is a speed-up of A_β with respect to g. Mathematics Subject Classification: 03F20, 03F30.     

Some results on the cartan matrix of a frobenius algebra

Let F be a field and A a Frobenius algebra over F. The Jacobson radical of A is denoted by J = J(A) and the kth socle of A by S _k (A). Let [Abar]=A/J ^k or A/S _k (A). This article gives new interesting relations between the Cartan matrix of A and that of [Abar]. From these results we prove that the Cartan matrix of A is diagonal if A/Soc(A) is a symmetric algebra. Let G be a finite group. If A is a block of F|G] with the above condition, then the Cartan matrix of A is (n), where n is the order of the defect group of A and the least integer such that Jⁿ (A)=0.     

Profinite Heyting Algebras

For a Heyting algebra A, we show that the following conditions are equivalent: (i) A is profinite; (ii) A is finitely approximable, complete, and completely join-prime generated; (iii) A is isomorphic to the Heyting algebra Up(X) of upsets of an image-finite poset X. We also show that A is isomorphic to its profinite completion iff A is finitely approximable, complete, and the kernel of every finite homomorphic image of A is a principal filter of A.      

Biring Theory and Its Galois Theory of Rings

Biring theory is about birings (A, P), that is, algops (A, P) of an associative algebra A and (A, A)-biring P acting on A via a morphism γ: P → Pres _F A from P to the terminal (A, A)-biring Pres _F A of preservations of A. (The word biring is used in a theory for a structure with unit, product, counit, coproduct subject to conditions of the theory.) Biring theory has its central simple theory and its Galois theory of rings. Its Galois birings are the reduced simple birings.     

On Characteristic Modules of Graded Quasi-hereditary Algebras

Abstract

Let G be a group and A a G-graded quasi-hereditary algebra. Then its characteristic module is proved to be G-gradable, i.e., it is isomorphic to a G-graded module as A-modules. This implies that the Ringel dual A′ of A admits a canonical G-grading which extends to the graded situation the typical equivalence between Δ-good and ?-good modules of A and A′, respectively. It follows some consequences: the derived category of finitely generated G-graded A-modules is equivalent to the derived category of finitely generated G-graded A'-modules; if G is finite, then the Ringel dual of the smash product A#G* is isomorphic to the smash product A'#G* of A' with G.     

π-Complementation in the Unitisation and Multiplication Algebras of a Semiprime Algebra

π-complemented algebras are defined as those (not necessarily associative or unital) algebras such that each annihilator ideal is complemented by other annihilator ideal. For a given semiprime algebra A, we discuss the π-complementation of the unitisation algebra A ¹ of A. Moreover, if in addition the multiplication algebra ?(A) of A is also semiprime, we study the π-complementation in the algebras ?(A) and ?^?(A) (the multiplication ideal of A). In associative setting, we prove that A is π-complemented if and only if ?^?(A) is π-complemented, and that A ¹ π-complemented if and only if ?(A) is π-complemented.     

On Codimension Growth of Finitely Generated Associative Algebras

LetAbe a PI-algebra over a fieldF. We study the asymptotic behavior of the sequence of codimensionsc_n(A) ofA. We show that ifAis finitely generated overFthenInv(A)=lim_n→∞ always exists and is an integer. We also obtain the following characterization of simple algebras:Ais finite dimensional central simple overFif and only ifInv(A)=dim=A.     

Cohomological Characterizations of Biprojective and Biflat Banach Algebras

 Let A be a biprojective Banach algebra, and let A-mod-A be the category of Banach A-bimodules. In this paper, for every given -mod-A, we compute all the cohomology groups . Furthermore, we give some cohomological characterizations of biprojective Banach algebras. In particular, we show that the following properties of a Banach algebra A are equivalent to its biprojectivity: (i) for all -mod -A; (ii) for all -mod-A; (iii) for all -mod-A. (Here and are, respectively, the Banach A-bimodules of left, right and double multipliers of X.) Further, if A is a biflat Banach algebra and -mod-A, we compute all the cohomology groups , where is the Banach A-bimodule dual to X. Also, we give cohomological characterizations of biflat Banach algebras. We prove that a Banach algebra A is biflat if and only if any of the following conditions is valid: (i’) for all -mod-A; (ii’) for all -mod-A; (iii’) for all -mod-A.     

Simple lie color algebras of weyl type

A class of graded simple associative algebras are constructed, and from them, simple Lie color algebras are obtained. The structure of these simple Lie color algebras is explicitly described. More precisely, for an (ε, Γ)-color-commutative associative algebraA with an identity element over a fieldF of characteristic not 2, and for a color-commutative subalgebraD of color-derivations ofA, denote byA[D] the associative subalgebra of End (A) generated byA (regarded as operators onA via left multiplication) andD. It is easily proved that, as an associative algebra,A[D] is Γ-graded simple if and only ifA is Γ-gradedD-simple. SupposeA is Γ-gradedD-simple. Then, (a)A[D] is a free leftA-module; (b) as a Lie color algebra, the subquotient [A[D],A[D]]/Z(A[D])∩[A[D],A[D]] is simple (except one minor case), whereZ(A[D]) is the color center ofA[D]. This work was supported by NSF of China, National Educational Department of China, Jiangsu Educational Committee, and Hundred Talents Program of Chinese Academy of Sciences. These authors were partially supported by Academy of Mathematics and System Sciences during their visit to this academy.     

Edge disjoint Steiner trees in graphs without large bridges

A set A of vertices of an undirected graph G is called k‐edge‐connected in G if for all pairs of distinct vertices a, b∈A, there exist k edge disjoint a, b‐paths in G. An A‐tree is a subtree of G containing A, and an A‐bridge is a subgraph B of G which is either formed by a single edge with both end vertices in A or formed by the set of edges incident with the vertices of some component of G ? A. It is proved that (i) if A is k·(? + 2)‐edge‐connected in G and every A‐bridge has at most ? vertices in V(G) ? A or at most ? + 2 vertices in A then there exist k edge disjoint A‐trees, and that (ii) if A is k‐edge‐connected in G and B is an A‐bridge such that B is a tree and every vertex in V(B) ? A has degree 3 then either A is k‐edge‐connected in G ? e for some e∈E(B) or A is (k ? 1)‐edge‐connected in G ? E(B). © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 188–198, 2009     

Generating primitive positive clones

Suppose is a set of operations on a finite set A. Define PPC() to be the smallest primitive positive clone on A containing . For any finite algebra A, let PPC#(A) be the smallest number n for which PPC(CloA) = PPC(Clo _n A). S. Burris and R. Willard [2] conjectured that PPC#(A) ≤|A| when CloA is a primitive positive clone and |A| > 2. In this paper, we look at how large PPC#(A) can be when special conditions are placed on the finite algebra A. We show that PPC#(A) ≤|A| holds when the variety generated by A is congruence distributive, Abelian, or decidable. We also show that PPC#(A) ≤|A| + 2 if A generates a congruence permutable variety and every subalgebra of A is the product of a congruence neutral algebra and an Abelian algebra. Furthermore, we give an example in which PPC#(A) ≥|A| - 1)² so that these results are not vacuous. Received August 30, 1999; accepted in final form April 4, 2000.