Characterisations of those separable C∗-algebras that have type I injective envelopes or W∗-algebra injective envelopes are presented. An operator system I is injective if for every inclusion E ⊂ F of operator systems each completely positive linear map ω : E → I has a completely positive extension to F . An injective envelope of an operator system E is an injective operator system I such that ...

The concepts of free modules, projective modules, injective modules, and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of projective and injective L-modules. In this paper, we give an alternate definition for injective L-modules and prove t...

We introduce the notion of integrality Grothendieck categories as a simultaneous generalization primeness noncommutative noetherian rings and locally schemes. Two different spaces associated to category yield respective definitions integrality, we prove equivalence these using Grothendieck-categorical version Gabriel's correspondence, which originally related indecomposable injective modules pr...

We assign a relational structure to any finite algebra in canonical way, using solution sets of equations, and we prove that this is polymorphism-homogeneous if only the itself polymorphism-homogeneous. show polymorphism-homogeneity also equivalent property algebraic (i.e., systems equations) are exactly those tuples closed under centralizer clone algebra. Furthermore, aforementioned properties...

Isbell and Semadeni [Trans. Amer. Math. Soc. 107 (1963)] proved that every infinite-dimensional $1$-injective Banach space contains a hyperplane is $(2+\varepsilon )$-injective for $\varepsilon \gt 0$, yet not $2$-injective, rem

LetR be a ring andM a rightR-module with S= End(MR). The moduleM is called almost principally quasi-injective (or APQ-injective for short) if, for any m∈M, there exists an S-submodule Xm of M such that lMrR(m) = Sm ⊕ Xm. The module M is called almost quasiprincipally injective (or AQP-injective for short) if, for any s∈ S, there exists a left ideal Xs of S such that lS(ker(s)) = Ss ⊕ Xs. In thi...