K-Groups of Trivial Extensions and Gluings of Abelian Categories

This paper focuses on the Ki-groups of two types extensions abelian categories, which are trivial extension and gluing categories. We prove that, under some conditions, a certian subcategory category is isomorphic to similar original category. Moreover, we show that left (right) categories direct sum As their applications, obtain results ring by bimodule (i∈N).