Hopf Dense Galois Extensions And Pseudo-Strongly Graded Algebras

Galois theory is always an important part of algebra,whose progress is indis-pensable for the development of algebra.In the study of Hopf algebra,Hopf Galois theory is also very important.In the decades,by the progress of understanding of Hopf Galois theory,its relevant theory and results are plentifully developed.Based on the theory of Hopf Galois theory,He-Van Oystaeyen-Zhang introduced the concept of Hopf dense Galois extensions over fields.Their study about Hopf dense Galois extensions gives many interesting results,including the truth of Aus-lander theorem under the condition of Hopf dense Galois extensions.This paper generalizes the concept of Hopf dense Galois extensions to commutative domain,and studies its properties under this consideration.Meanwhile,based on the theory of Hopf dense Galois extension,He-Van Oystaeyen-Zhang introduced the concept of densely graded algebras.Densely graded algebras can be regarded as a generalization of strongly graded alge-bras,and they proved the Dade's theorem of densely graded algebras.Motived by the study of densely graded algebras,this paper introduces the concept of pseudo-strongly graded algebras.Pseudo-strongly graded algebras have similar properties with densely graded algebras.After the proof of Dade's theorem of pseudo-strongly graded algebras,this paper studies graded algebras with similar properties.