The classes of FP -injective and weakly quasi-Frobenius rings are investigated. The properties for both classes of rings are closely linked with embedding of finitely presented modules in fp-flat and free modules respectively. Using these properties, we characterize the classes of coherent CF and FGF-rings. Moreover, it is proved that the group ring R(G) is FP -injective (weakly quasi-Frobenius...

Preenvelopes of acts over a monoid are defined by analogy with Enochs’ definition preenvelopes modules. Provided that it is closed for pure subacts, class shown to be preenveloping precisely when under direct products. Examples such classes include absolutely pure, weakly f-injective and p-injective acts.

Existence of superdecomposable pure-injective modules reflects complexity in the category of finite-dimensional representations. We describe the relation in terms of pointed modules. We present methods for producing superdecomposable pure-injectives and give some details of recent work of Harland doing this in the context of tubular algebras. 2010 Mathematics Subject Classification. Primary 16G...

Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relation x −→ x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and id...

Using only basic topological properties of real algebraic sets and regular morphisms we show that any injective regular self-mapping of a real algebraic set is surjective. Then we show that injective morphisms between germs of real algebraic sets define a partial order on the equivalence classes of these germs divided by continuous semi-algebraic homeomorphisms. We use this observation to deduc...

For graphs G and H , a homomorphism from G to H is a function φ : V (G) → V (H), which maps vertices adjacent in G to adjacent vertices of H . A homomorphism is locally injective if no two vertices with a common neighbor are mapped to a single vertex in H . Many cases of graph homomorphism and locally injective graph homomorphism are NPcomplete, so there is little hope to design polynomial-time...

A brief comment about items (3) and (4). If H is a subgroup of G the inclusion map is an injective homomorphism from H to G . On the other hand, the image of an injective homomorphism α : H −→ G is a subgroup that is isomorphic to H . So the study of injective homomorphisms is (up to isomorphism) the study of subgroups. After studying subgroups, we defined normal subgroup and showed several equ...

Several possible definitions of local injectivity for a homomorphism an oriented graph G to H are considered. In each case, we determine the complexity deciding whether there exists such when is given and fixed tournament on three or fewer vertices. Each definition leads locally-injective colouring problem. A dichotomy theorem proved in case.

As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square I2 in R2 (or more generally, of the cube I in R) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of I2 (or I). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is suffic...