Several new characterizations of Banach spaces containing a subspace isomorphic to , are obtained. These are applied to the question of when 1 embeds in the injective tensor product of two Banach spaces. Notations and terminology. All Banach spaces are taken as infinite dimensional, “subspace” means “closed linear subspace,” “operator” means “bounded linear operator.” If W is a subset of a Bana...

Let R be a ring. A right R-module is said to C-flat if the kernel of any epimorphism B → C-pure in B, i.e. induced map Hom(C,B) Hom(C,A) surjective for cyclic C. Projective modules are and weakly-flat neat-flat. In this article, it discussed connections between C-flat, weakly-flat, neat-flat singly flat modules. It shown that coincide with singly-projective over arbitrary rings. Next, several c...

Proof. The idempotents correspond to the Borel sets modulo sets of first category. Since, in addition, the idempotents generate B(X), B(X) is an AW* and hence an injective algebra. The natural map U of C(X) into B(X) induced by the inclusion map is clearly a homomorphism. It is one-one since continuous functions which are not identically equal must differ on a set of second category. To complet...

We define and studyco-Noetherian dimension of rings for which the injective envelopeof simple modules have finite Krull-dimension. This  is a Moritainvariant dimension that measures how far the ring is from beingco-Noetherian. The co-Noetherian dimension of certain rings,including commutative rings, are determined. It is shown that the class ${mathcal W}_n$ of rings with co-Noetherian dimension...

We consider projectivity and injectivity of Hilbert C*-modules in the categories of Hilbert C*-(bi-)modules over a fixed C*-algebra of coefficients (and another fixed C*-algebra represented as bounded module operators) and bounded (bi-)module morphisms, either necessarily adjointable or arbitrary ones. As a consequence of these investigations, we obtain a set of equivalent conditions characteri...

In this paper, we show that if an injective map  on symmetric matrices   n S C satisfies then           , , n ABA A B A A B S        C ,   Φ t f A SA S   for all   n A S C  , where f is an injective homomorphism on , is a complex orthogonal matrix and C S f A is the image of A under f applied entrywise.

Let $mathcal{X}$ be a class of $R$-modules‎. ‎In this paper‎, ‎we investigate ;$mathcal{X}$-injective (projective) and DG-$mathcal{X}$-injective (projective) complexes which are generalizations of injective (projective) and DG-injecti‎‎ve (projective) complexes‎. ‎We prove that some known results can be extended to the class of ;$mathcal{X}$-injective (projective) and DG-$mathcal{X}$-injective ...

let $mathcal{x}$ be a class of $r$-modules‎. ‎in this paper‎, ‎we investigate ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective (projective) complexes which are generalizations of injective (projective) and dg-injecti‎‎ve (projective) complexes‎. ‎we prove that some known results can be extended to the class of ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective ...

Following Perelman’s solution to the Geometrisation Conjecture, a ‘generic’ closed 3-manifold is known to admit a hyperbolic structure. However, our understanding of closed hyperbolic 3-manifolds is far from complete. In particular, the notorious Virtually Haken Conjecture remains unresolved. This proposes that every closed hyperbolic 3-manifold has a finite cover that contains a closed embedde...