We study the issue of purity (as a completely positive linear map) for identity maps on operators systems and their isometric embeddings into C$^*$-envelopes injective envelopes. Our most general result states that canonical embedding an operator system $\mathcal R$ its envelope I(\mathcal R)$ is pure if only C$^*$-envelope prime C$^*$-algebra. To prove this, we also show map any AW$^*$-factor ...

For a one-dimensional local domain CM constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of CM is described using these residue maps. Injective hulls of a given module are all isomorphic. For this reason, people often speak of the injective hull to indicate its “uniqueness”. However, isomorphisms between these injective hulls are not canonical. In fa...

This lectures were given by Florian Enescu at the mini-course on classical problems in commutative algebra held at University of Utah, June 2004. The references listed were used extensively in preparing these notes and the author makes no claim of originality. Moreover, he encourages the reader to consult these references for more details and many more results that had to be omitted due to time...

For a unitary right module M , there are two known partitions of simple modules in the category σ[M ]: the first one divides them into M -injective modules and M -small modules, while the second one divides them into M -projective modules and M -singular modules. We study inclusions between the first two and the last two classes of simple modules in terms of some associated radicals and proper ...

α1 α2 α3 α4 inModR. If the rows are exact, then the following statements hold. (1.3.a) If γ2 and γ4 are injective and γ1 is surjective, then γ3 is injective. (1.3.b) If γ2 and γ4 are surjective and γ5 is injective, then γ3 is surjective. Proof. First we prove (1.3.a). So suppose that γ2 and γ4 are injective and γ1 is surjective. Start with m3 ∈ M3 with the property that γ3(m3) = 0. The goal is ...

For a one-dimensional local domain CM constructed by Akizuki, we find residue maps which give rise to a local duality. The completion of CM is described using these residue maps. Injective hulls of a given module are all isomorphic. For this reason, people often speak of the injective hull to indicate its “uniqueness”. However, isomorphisms between these injective hulls are not canonical. In fa...

Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:math.CT/0804.0326] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadic...

Let M be a closed oriented surface of genus g ≥ 1, let Bn(M) be the braid group of M on n strings, and let SBn(M) be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map η : SBn(M) → Z[Bn(M)], introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective. AMS Subject Classification: Primary 20F36; Seconda...

In this paper, we study the group Pext(C, A) for locally compact abelian (LCA) groups A and C. Sufficient conditions are established for Pext(C, A) to coincide with the first Ulm subgroup of Ext(C, A). Some structural information on pure injectives in the category of LCA groups is obtained. Letting C denote the class of LCA groups which can be written as the topological direct sum of a compactl...

An injective coloring of a graph is a vertex labeling such that two vertices sharing a common neighbor get different labels. In this work we introduce and study what we call additive colorings. An injective coloring c : V (G) → Z of a graph G is an additive coloring if for every uv, vw in E(G), c(u)+ c(w) = 2c(v). The smallest integer k such that an injective (resp. additive) coloring of a give...