N-Frobenius Pairs

Based on the Frobenius pair,this paper introduces the definition of n-Frobenius pair in different categories,and discusses the related properties.In part 1,we mainly give the definition of n-Frobenius pair in the abelian category and the relationship between n-Frobenius pair and Frobenius pair,study the properties of n-Frobenius pair in the abelian category,and discuss the relationship between left(right)nFrobenius pair and right(left)n-cotorsion pair in the abelian category.In addition,the concept of special left(right)n-Frobenius pair in the abelian category and its equivalent characterizations are also given.Finally,some examples of left(right)n-Frobenius pairs and special left(right)n-Frobenius pairs in the abelian category are given through Gorenstein modules,and some characterizations of rings are also given.In part 2,the definition of n-Frobenius pairs in the extriangulated category is introduced,the relationship between n-Frobenius pairs and Frobenius pairs is discussed,and the relationship between the left(right)n-Frobenius pair and the right(left)n-cotorsion pair in the extriangulated category is checked.Moreover,the concept of special left(right)n-Frobenius pairs in the extriangulatde category is introduced and some equivalent characterizations are given.Finally,the equivalence relationship between the special n-Frobenius pair and the(n+1)-cluster tilting subcategory on the extriangulated category with enough projective objects and enough injective objects is showed.