Let (R, m, K) be a Gorenstein local ring of prime characteristic p > 0, and let x1, . . . , xn be a system of parameters. Let It = (x1, . . . , x t n)R for all t ≥ 1, and let u ∈ R represent a socle generator in R/I, where I = I1 = (x1, . . . , xn)R. Let y = x1 · · · xn. We have seen that E = lim −→ t R/It is an injective hull of K = R/m over R, where the map R/It → R/It+1 is induced by multipl...

We study identifiability for bilinear inverse problems under sparsity and subspace constraints. We show that, up to a global scaling ambiguity, almost all such maps are injective on the set of pairs of sparse vectors if the number of measurements m exceeds 2(s1+ s2)− 2, where s1 and s2 denote the sparsity of the two input vectors, and injective on the set of pairs of vectors lying in known subs...

In this paper we show that a direct decomposition of modules M N, with N homologically independent to the inJective hull of H, is a CS-module if and only if N is injective relative to H and both of M and N are CS-modules. As an application, we prove that a direct sum of a non-singular semisimple module and a quasi-continuous module with zero socle is quasi-continuous. This result is known for q...

1.1. Injective resolutions. Let C be an abelian category. An object I ∈ C is injective if the functor Hom(−, I) is exact. An injective resolution of an object A ∈ C is an exact sequence 0→ A→ I → I → . . . where I• are injective. We say C has enough injectives if every object has an injective resolution. It is easy to see that this is equivalent to saying every object can be embedded in an inje...

A set-valuation of a graph G = (V, E) assigns to the vertices or edges of G elements of the power set X 2 of a given nonempty set X subject to certain conditions and set-valuations have a variety of origins. Acharya defined a set-indexer of G to be an injective set-valuation X G V f 2 ) ( : → such that the induced set-valuation ) ( : G E f ⊕ X 2 → on the edges of G defined by ) ( ), ( ) ( ) ( G...

We determine the minimal lower bound n, with $$n \geqslant 1$$ , where n-th power of radical module category a representation-finite cluster tilted algebra vanishes. give such in terms number vertices underline quiver. Consequently, we get nilpotency index for self-injective algebras. also study non-zero composition m, $$m 2$$ irreducible morphisms between indecomposable modules algebras lying ...

It is no exaggeration to say that the theory of separably injective spaces is quite different from that of injective spaces. In this chapter we will explain why. Indeed, we will enter now in the main topic of the monograph, namely, separably injective spaces and their “universal” version. After giving the main definitions and taking a look at the first natural examples one encounters, we presen...