let r be a commutative noetherian ring. we study the behavior of injectiveand at dimension of r-modules under the functors homr(-,-) and -×r-.

In this article, we show that the localization of an extriangulated category by a multiplicative system satisfying mild assumptions can be equipped with natural, universal structure category. This construction unifies Serre quotient abelian categories and Verdier triangulated categories. Indeed give such for bit wider class morphisms, so it covers several other localizations appeared in literat...