Injective hulls for ordered algebras

We study the injectivity of ordered algebras with respect to the class of homomorphisms that are order-embeddings and one more class of morphisms. We do this in the category where morphisms need not be homomorphisms, but satisfy a condition which is weaker than operation-preservation. In this setting, the injective objects turn out to be precisely sup-algebras. We also show how to construct injective hulls of ordered algebras satisfying certain conditions.     

Injective and projective regular double Stone algebras

We characterize the injective and projective regular double Stone algebras, and describe those regular double Stone algebras which are also projective in the category of double Stone algebras.Presented by Bjarni Jónsson.     

Symmetrische Algebren mit injektivem Zentrum

The group ring of a finite group G over a field K is a symmetric algebra and hence an injective ring. If K is of characteristic zero, KG is semisimple. Thus the center of KG is injective. We proof in this paper: If K is of characteristic p>0, then the center of KG is injective, if and only if G is p-nilpotent and the p-Sylow-subgroups of G are abelian. We determine also the structure of these group rings. This will be done more generally for symmetric algebras which have an injective center and split over its Jacobson-radical.     

Gröbner–Shirshov Bases for Universal Enveloping Conformal Algebras of Simple Conformal Lie Superalgebras of Type WN

For simple conformal Lie superalgebras of type W_N, Gröbner–Shirshov bases of their universal enveloping associative conformal algebras are found. The universal enveloping algebras considered correspond to a minimal locality function for which there is an injective embedding.     

Purity in algebra

We investigate pure monomorphisms and pure epimorphisms in varieties of universal algebras with applications to equationally compact (= pure injective) algebras, pure projective algebras and perfect varieties. Received December 3, 2004; accepted in final form November 19, 2005. The second author was supported by the Grant Agency of the Czech Republic under the grant 201/02/0148. The hospitality of the Catholic University Louvain is gratefully acknowledged.     

Pure Injective Indecomposable Modules over 1-Domestic String Algebras

We classify indecomposable pure injective modules over 1-domestic string algebras verifying Ringel’s conjecture on their structure.     

Zur Existenz freier Algebren einer r-dimensionalen Theorie

In [2] Linton introduced the notion of theory in such a way that he was able to indicate explicitly the leftadjoint of the underlying functor from the category of algebras over the theory in the sets. His method defines, in the case of the restriction on a r-ary theory [2] or a algebraische Theorie der Dimension r [3] (r an infinite cardinal number), the leftadjoint only for sets with cardinality less than r. In this paper it will be purely categorically shown what from the point of view of universal algebra is wellknown, that the remaining free algebras over a given r-ary theory can be obtained as injective limits of free algebras over sets with cardinality less than r. The proof stems from the following two observations: first that all sets are injective limits of the r-directed system of their subsets with cardinality less than r and secondly that for all sets m with cardinality less than r the Hom-functor preserves injective limits of r-directed systems of sets. In an appendix the endofunctors of the category of sets which preserve injective limits of r-directed systems of sets are characterized by the same property that the functor-part of a triple in the sets has by definition, if r is the rank of the triple [4].

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Supported by the National Research Council of Canada     

Injective modules and representational repleteness

The purpose of this paper is to investigate the relationship between the associated primes of non-zero factors of a uniform injective module over a Noetherian ring and the right clique of the assassinator, when the clique satisfies the second layer condition. We introduce the terminology “representationally replete” to refer to primes where every linked prime occurs as such an associated prime and we study this property in several classes of rings including enveloping algebras of solvable Lie algebras and finite dimensional algebras     

A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space

In 1975 A. Connes proved the fundamental result that injective factors on a separable Hilbert space are hyperfinite. In this paper a new proof of this result is presented in which the most technical parts of Connes proof are avoided. Particularly the proof does not rely on automorphism group theory. The starting point in this approach is Wassermann's simple proof of injective ? semidiscrete together with Choi and Effros' characterization of semidiscrete von Neumann algebras as those von Neumann algebras N for which the identity map on N has an approximate completely positive factorization through n × n-matrices.     

Global Weyl modules for the twisted loop algebra

We define global Weyl modules for twisted loop algebras and analyze their highest weight spaces, which are in fact isomorphic to Laurent polynomial rings in finitely many variables. We are able to show that the global Weyl module is a free module of finite rank over these rings. Furthermore we prove, that there exist injective maps from the global Weyl modules for twisted loop algebras into a direct sum of global Weyl modules for untwisted loop algebras. Relations between local Weyl modules for twisted and untwisted generalized current algebras are known; we provide for the first time a relation on global Weyl modules.     

Weakly Regular and Self-Injective Leavitt Path Algebras Over Arbitrary Graphs

We characterize the Leavitt path algebras over arbitrary graphs which are weakly regular rings as well as those which are self-injective. In order to reach our goals we extend and prove several results on projective, injective and flat modules over Leavitt path algebras and, more generally, over (not necessarily unital) rings with local units.     

Natural dualities for varieties of BL-algebras

BL-algebras are the Lindenbaum algebras for Hájek's Basic Logic, just as Boolean algebras correspond to the classical propositional calculus. The finite totally ordered BL-algebras are ordinal sums of MV-chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite BL-chain, and we use it to describe the injective and the weak injective members of these classes. The preliminary research for this paper was carried out while the second author was visiting Salerno University. The second author would like to thank the first author and Salerno University for their hospitality. The second author acknowledges partial supports from Salerno University and from the belgian Fonds National de la Recherche Scientifique.     

The Cartan matrix of a centralizer algebra

The centralizer algebra of a matrix consists of those matrices that commute with it. We investigate the basic representation-theoretic invariants of centralizer algebras, namely their radicals, projective indecomposable modules, injective indecomposable modules, simple modules and Cartan matrices. With the help of our Cartan matrix calculations we determine their global dimensions. Many of these algebras are of infinite global dimension.     

W~*-三元算子环的摄动

本文证明了:如果两个W~*-三元算子环V和W的cb距离d_(cb)(V,W)很小的时候,其连接冯·诺依曼代数之间的距离也很小.还证明了:和内射的W~*-三元算子环靠的很近的W~*-三元算子环也是内射的.对具有r性质和McDuff性质的W~*-三元算子环,类似的结论也成立.     

Noninvertibility preservers on Banach algebras

It is proved that a linear surjection Ф: A → B, which preserves noninvertibility between semisimple, unital, complex Banach algebras, is automatically injective.     

Algebras determined by their supports

In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander–Reiten components of an ada algebra which is not quasi-tilted, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes.     

Igusa–Todorov functions for Artin algebras

In this paper we study the behavior of the Igusa–Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the ?-dimension and ψ-dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite ?-dimension and finite ψ-dimension.     

Cohomological reduction by split pairs

A new method is developed to compare cohomology in module categories of different rings. This method does in general not produce isomorphisms, but surjective (or injective) maps between extension groups of modules over the two rings involved. Applications of this method are given to abstract problems—we recover and extend results on the strong no loops conjecture—and to algebras naturally coming up in invariant theory—we relate the cohomology of Brauer algebras with that of various symmetric groups.     

关于BCI—代数伴随半群中的可逆元

给出了ＢＣＩ－代数的伴随半群中可逆元的刻划，说明ＢＣＩ－代数的ｐ－半单闭理想与其伴随半群的子群之间有一一对应关系，并证明最大的ｐ－半单闭理想是根理想。