Combination Of Algebraic Nature

Because Wedderburn-Malcev Theorem plays an important role in algebra, manymathematical scientists have done research on it, As an application of cohomology the-ory,someone state Wedderburn-Malcev Theory for associative algebras,and someone statea generalization of Wedderburn-Malcev Theorem,for instance, D.Stefan and F.Van Oys-taeyen state a generalization of Wedderburn-Malcev Theorem for comodule algebras andits dual result.In chapter 1,we first give some related knowledge and all other different forms ofWedderburn-Malcev Theorem,in the following we state the Wedderburn-Malcev Theoremfor module algebra and for comodule coalgebra.Theroem Let H be a finite dimension semisimple Hopf algebra such that there is anonzero left integralλ∈H_ad,Let A be a finite dimension module algebra over H such thatJ,the Jacobson radical of A,is a submodule of A and A/J(A) is separable.Then there is asubmodule algebra B in A such that: A=B⊕JThis decomposition as direct sum of H module.Theroem Let H be a finite dimension cosemisimple Hopf algebra.suppose thata nonzero integral of H^* satisfiesâ–³(t)=Σt₂(?)S²(t₁).Let C be a right H-comodulecoalgebra. If the coradical C₀ of C is coseparable and H^*-stable,then there is an H^*-linearcoalgebra projection:Câ†'C₀.In chapter 2,we state complexity of Crossed product,that is the following theorem:Theroem suppose H is a finite dimension Hopf algebra and A is a H-modulealgebra,if H and H^* are semisimple,then we have the following result: C(A#_σH)=C(A)In chapter 3,we give a proposition and a theorem,we replace algebraically closed fieldwith an ordinary field,thai is:Proposition Let A and B be two finite dimension simple algebras over field K withcharacteristic P is satisfied P(?)(dim_D(A))^1/2 [D:k],Then,for an A-B bimodule M. theisomorphism of B-A bimodules Hom_A(M, A)≌Hom_B(M, B) holds. Lemma Assume A is a finite dimension simple algebra over field K with charac-teristic P is satisfied P (?)(dim_D(A))^1/2 [D:k],Then, for anyα≠0 in A,there holds thatt(αA)≠0.In chapter 4,we replace S is a group with G is a Clifford semigroup,and we have thesimilar result,that is:Theroem Suppose S is a Clifford semigroup.For x, y∈S,letR₁ (x, y)=(x^-1, xyx),R₂(x, y)=(x, xyx^-1), R₃(x, y)=(x^-1x, xy), R₁(x, y)=(xx^-1, xy).Then R_i(i=1, 2, 3, 4)are solutions of the QYBE.