A Class Extension Of An Algebra And Its Category Of Modules

Algebra extension is a process of forming a kind of new algebra got ac-cording to certain rule from a known algebra. Algebra extension and relevantnature of extension algebra have been the basic problem in the study of al-gebra. Our main research antithesis of academic dissertation is the extensionof algebra T(A, J), repetitive algebra (?)(A, J) and the relation between thecategories of the module of two algebra .The extension algebra T come fromthe half-direct product of an algebra A and D=Hom(A/J, k) and J is theideal of A. Algebra (?)(A, J) is repetitive algebra of T(A, J). This academicdissertation is divided into four chapters altogether.Chapter one, we carry on the introduction to the research direction andthe trends of the development related to thesis, and sum up the groundworkof this text.Chapter two, we study the extension of algebra T(A, J) and the repetitivealgebra (?)(A, J), we also describe the categories of module of the extensionof algebra T(A,J) and algebra (?)(A/J). In addtition, we give indecompos-able projective T(A, J) module and (?)(A, J)-mod, indecomposable injectiveT(A, J)-mod and (?)(A, J)-mod, single T(A, J)-mod.Chapter three, we pay attention to the relationship between the categoriesT(A, J)-mod and (?)(A, J)-Mod. We denote by (?)(A, J)-Mod the category of(?)(A, J)-Mod X such that X_n has finite k-dimension for all integers n and by(?)(A, J)-mod the full subcategory of (?)(A, J)-Mod of (?)(A, J)-mod X suchthat (?)_n X_n has k-dimension.The automorphism v of (?)(A, J) induce auto-morphisms of the categories (?)(A, J)-Mod and (?)(A, J)-mod. again denoted v:X(?) vX where(vX)_n=X_n-1 and (vf)_n=f_n-1 for all integers n. The sub-category of (?)(A, J)-Mod consisiting of v-invariant modules and v-invariantmorphisms is denoted (?)(A, J)-Mod. We proof that the categories T(A, J)-mod and (?)(A,J)-Mod are equivalent.We also proof that there is a coveringfunctor from (?)(A, J)-mod to T(A, J)-mod.Chapter four, we study the relationship between the trivial extension al-gebra T(A/J) and extension algebra T(A, J).