Annihilator Primes and Foundation Primes

ABSTRACT

Let R be a Noetherian ring and M a finitely generated R -module. In this article, we introduce the set of prime ideals Fnd  M , the foundation primes of M . Using the fact that this set is nicely organized by foundation levels, we present an approach to the problem of understanding Annspec  M , the annihilator primes of M , via Fnd  M . We show: (1) Fnd  M is a finite set containing Annspec  M . Further, suppose that moreover every ideal of R has a centralizing sequence of generators; now, Annspec  M is equal to the set Ass  M of associated primes of M. Then: (2) For an arbitrary P  ∈ Fnd  M , P  ∈ Annspec  M if and only if there is no Q  ∈ Annspec  M such that P contains Q , and at the same time, the minimal foundation level on which appears P is greater than the minimal foundation level on which appears Q .     

The Converse of Schur's Lemma in Noetherian Rings and Group Algebras

ABSTRACT

If M is a simple module over a ring R, then, by Schur's Lemma, its endomorphism ring is a division ring. However, the converse of this property, which we called the CSL property, does not hold in general. The object of this article is to study this converse for a few classes of rings: left Noetherian rings, V-rings and group algebras. First, we establish that a left Noetherian ring R is a CSL ring if and only if a ring R is left–artinian and primary decomposable. Secondly, we prove that a left semiartinian V-ring is CSL. At last, we study the CSL property in group algebra K [ G ] where K a field algebraically closed of characteristic p and G is a finite group of order divisible by p. Our main contribution is that K [ G ] is a CSL ring if and only if Gbf = HP where H is a normal p′-subgroup and bfP a Sylow bfp-subgroup of bfG. In this case, K [ G ] is primary decomposable.     

Purity and Self-Small Groups

Let A be an abelian group. A group B is A-solvable if the natural map Hom(A, B) ?  _E(A) A → B is an isomorphism. We study pure subgroups of A-solvable groups for a self-small group A of finite torsion-free rank. Particular attention is given to the case that A is in , the class of self-small mixed groups G with G/tG? ? ⁿ for some n < ω. We obtain a new characterization of the elements of , and demonstrate that  differs in various ways from the class  ? of torsion-free abelian groups of finite rank despite the fact that the quasi-category ?  is dual to a full subcategory of ?  ?.     

Monotone clones and congruence modularity

We investigate the relationship between the local shape of an ordered set P=(P; ) and the congruence-modularity of the variety V generated by an algebra A=(P; F) each of whose operations is order-preserving with respect to P. For example, if V is k-permutable (k2) then P is an antichain; if P is both up and down directed and V is congruence-modular, then V is congruence-distributive; if A is a dual discriminator algebra, then either P is an antichain or a two-element chain. We also give a useful necessary condition on P for V to be congruence-modular. Finally a class of ordered sets called braids is introduced and it is shown that if P is a braid of length 1, in particular if P is a crown, then the variety V is not congruence-modular.     

Multivariate normal distributions,Fisher information and matrix inequalities

Using appropriately parameterized families of multivariate normal distributions and basic properties of the Fisher information matrix for normal random vectors, we provide statistical proofs of the monotonicity of the matrix function A ^-1 in the class of positive definite Hermitian matrices. Similarly, we prove that A ¹¹ &lt; A ^-111, where A ₁₁ is the principal submatrix of A and A ¹¹ is the corresponding submatrix of A ^-1. These results in turn lead to statistical proofs that the the matrix function A ^-1 is convex in the class of positive definite Hermitian matrices and that A ² is convex in the class of all Hermitian matrices. (These results are based on the Loewner ordering of Hermitian matrices, under which A &lt; B if A - B is non-negative definite.) The proofs demonstrate that the Fisher information matrix, a fundamental concept of statistics, deserves attention from a purely mathematical point of view.     

Symplectic algebras and poisson algebras

Abstract

A ring R is called left P-injective if for every a ∈ R, aR = r(l(a)) where l( ? ) and r( ? ) denote left and right annihilators respectively. The ring R is called left GP-injective if for any 0 ≠ a ∈ R, there exists n > 0 such that a ⁿ  ≠ 0 and a ⁿ R = r(l(a ⁿ )). As a response to an open question on GP -injective rings, an example of a left GP-injective ring which is not left P-injective is given. It is also proved here that a ring R is left FP -injective if and only if every matrix ring 𝕄 _n (R) is left GP-injective.     

Varieties of restriction semigroups and varieties of categories

The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, ^?1) by forgetting the inverse operation and retaining the two operations x⁺ = xx^?1 and x* = x^?1x. The subvariety B of strict restriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals [B₂, B₂M = B] and [B₀, B₀M]. Here, B₂ and B₀ are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval [B₂, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson's major theorems have natural interpretations and application to the interval [B₂, B] and, with modification, to the interval [B₀, B₀M] that lies below it. Further exploration may lead to applications in the reverse direction.     

Orthogonal Dimensions and Ring Extensions

Let , ? be classes of modules, we introduce and characterize orthogonal dimensions of modules and rings by the orthogonal classes of modules  ^⊥, ^⊥ ?, and  ^T . For an almost excellent extension S ≥ R, we give the criteria to test some identities of orthogonal dimensions of modules and rings. As corollaries, some known results are obtained or extended.     

Some remarks on dualisability and endodualisability

The fundamental problem of dualisability and the particular problem of endodualisability are discussed. It is proved tha every finite generating algebra of a quasi-variety generated by a finite dualisable algebra D is also dualisable. The corresponding result for endodualisability is true when D is subdirectly irreducible. Under special conditions, it is also proved that a finite algebra M is endodualisable if and only if any finite power M ⁿ of M is endodualisable. Received January 27, 1999; accepted in final form September 17, 1999.     

On some non-obvious connections between graphs and unary partial algebras

In the present paper we generalize a few algebraic concepts to graphs. Applying this graph language we solve some problems on subalgebra lattices of unary partial algebras. In this paper three such problems are solved, other will be solved in papers [Pió I], [Pió II], [Pió III], [Pió IV]. More precisely, in the present paper first another proof of the following algebraic result from [Bar1] is given: for two unary partial algebras A and B, their weak subalgebra lattices are isomorphic if and only if their graphs G*(A) and G*(B) are isomorphic. Secondly, it is shown that for two unary partial algebras A and B if their digraphs G(A) and G(B) are isomorphic, then their (weak, relative, strong) subalgebra lattices are also isomorphic. Thirdly, we characterize pairs , where A is a unary partial algebra and L is a lattice such that the weak subalgebra lattice of A is isomorphic to L.     

A Property About Minimum Edge- and Minimum Clique-Cover of a Graph

 Let G be a graph with n vertices, and denote as γ(G) (as θ(G)) the cardinality of a minimum edge cover (of a minimum clique cover) of G. Let E (let C) be the edge-vertex (the clique-vertex) incidence matrix of G; write then P(E)={x∈ℜ ⁿ :Ex≤1,x≥0}, P(C)={x∈ℜ ⁿ :Cx≤1,x≥0}, α _E (G)=max{1 ^T x subject to x∈P(E)}, and α _C (G)= max{1 ^T x subject to x∈P(C)}. In this paper we prove that if α _E (G)=α _C (G), then γ(G)=θ(G). Received: May 20, 1998?Final version received: April 12, 1999     

Dimension,Graph and Hypergraph Coloring

There is a natural way to associate with a poset P a hypergraph H _P, called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of H _P. The ordinary graph G _P of incomparable pairs determined by the edges in H _P of size 2 can have chromatic number substantially less than H _P. We give a new proof of the fact that the dimension of P is 2 if and only if G _P is bipartite. We also show that for each t 2, there exists a poset P _t for which the chromatic number of the graph of incomparable pairs of P _t is at most 3 t – 4, but the dimension of P _t is at least (3 / 2) ^{t – 1}. However, it is not known whether there is a function f: NN so that if P is a poset and the graph of incomparable pairs has chromatic number at most t, then the dimension of P is at most f(t).     

Quasi-Frobenius Corings and Quasi-Frobenius Extensions

In this article, the notions of a Frobenius pair of functors and Frobenius corings are generalized to an l-QF pair of functors and l-QF corings. We prove that an extension ι:B → A is left quasi-Frobenius if and only if (F ₁,G ₁) is an l-QF pair of functors, where F ₁: _A ? →  _B ? is the restriction of scalars functors, and G ₁ = A? _{B ^?} : _B ? →  _A ? is the induction functor. For an A-coring , we prove that  is an l-QF coring if and only if A → ? is an l-QF extension and _A  is a finitely generated projective modules if and only if (G ₂,F ₂) is an l-QF pair of functors, where G ₂ =  ? _{A ^?} : _A ? →  ? is the induction functor, F ₂:  ? →  _A ? is the forgetful functor, the result of Brzezinski is generalized.     

Morita duality and graded rings

ABSTRACT

We find representatives of all the equivalence classes of simple root systems (or r.e.s. for brevity) for the complex reflection groups G ₁₂ , G ₂₄ , G ₂₅ and G ₂₆ . Then we give representatives of all the congruence classes of (essential) presentations (or r.c.p. (r.c.e.p.) for brevity) for these groups by generators and relations. The method used in the paper is applicable to any finite (complex) reflection groups.     

Almost distributive sublattices and congruence heredity

A congruence lattice L of an algebra A is hereditary if every 0-1 sublattice of L is the congruence lattice of an algebra on A. Suppose that L is a finite lattice obtained from a distributive lattice by doubling a convex subset. We prove that every congruence lattice of a finite algebra isomorphic to L is hereditary. Presented by E. W. Kiss. Received July 18, 2005; accepted in final form April 2, 2006.     

The r-cubical Lattice and a Generalization of the cd-index

In this paper we generalize thecd-index of the cubical lattice to anr-cd-index, which we denote byΨ(r). The coefficients ofΨ(r) enumerate augmented Andrér-signed permutations, a generalization of Purtill's work relating thecd-index of the cubical lattice and signed André permutations. As an application we use ther-cd-index to determine that the extremal configuration which maximizes the Möbius function of arbitrary rank selections, where all ther_i's are greater than one, is the odd alternating ranks, {1, 3, 5, ...}.     

Interpolation by holomorphic automorphisms and embeddings in Cn

Let n > 1 and let C ⁿ denote the complex n-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings F:C ⁿ →C ⁿ and for holomorphic automorphisms of C ⁿ on discrete subsets of C ⁿ.We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into C ⁿ.For each closed complex submanifold (or subvariety) M ⊂ C ⁿ of complex dimension m < n we construct a domain Ω ⊂C ⁿ containing M and a biholomorphic map F: Ω → C ⁿ onto C ⁿ with J F ≡ 1such that F(M) intersects the image of any nondegenerate entire map G:C ^n−m →C ⁿ at infinitely many points. If m = n − 1, we construct F as above such that C ⁿ ∖F(M) is hyperbolic. In particular, for each m ≥ 1we construct proper holomorphic embeddings F:C ^m →C ^m−1 such that the complement C ^m+1 ∖F(C ^m )is hyperbolic.     

Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs

We consider both facial reduction, FR, and symmetry reduction, SR, techniques for semidefinite programming, SDP. We show that the two together fit surprisingly well in an alternating direction method of multipliers, ADMM, approach. In fact, this approach allows for simply adding on nonnegativity constraints, and solving the doubly nonnegative, DNN , relaxation of many classes of hard combinatorial problems. We also show that the singularity degree remains the same after SR, and that the DNN relaxations considered here have singularity degree one, that is reduced to zero after FR. The combination of FR and SR leads to a significant improvement in both numerical stability and running time for both the ADMM and interior point approaches. We test our method on various DNN relaxations of hard combinatorial problems including quadratic assignment problems with sizes of more than \(n=500\). This translates to a semidefinite constraint of order 250, 000 and \(625\times 10^8\) nonnegative constrained variables, before applying the reduction techniques.