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 ...

Let R be a ring and n a fixed positive integer, we investigate the properties of n-strongly Gorenstein projective, injective and flat modules. Using the homological theory , we prove that the tensor product of an n-strongly Gorenstein projective (flat) right R -module and projective (flat) left R-module is also n-strongly Gorenstein projective (flat). Let R be a coherent ring ,we prove that the...