The objective of writing this manuscript is to apply the concept of fuzzy set on some basic Hopf algebraic structures. In this manuscript, the novel concepts of fuzzy Hopf subalgebra, fuzzy Hopf ideal, and fuzzy -submodule are proposed. Some properties of these concepts are discussed, and some significant results are also proved in it. The advantages of the proposed work are also studied in it. The application of the proposed work is also discussed in it.
In this note, Cline's formula for generalized Drazin-Riesz inverses is proved. We prove that if A, D ∈ B(X, Y) and B, C ∈ B(Y, X) are such that ACD = DBD and DBA = ACA, then AC is generalized Drazin-Riesz invertible if and only if BD is generalized Drazin-Riesz invertible, and that, in such a case, if S is a generalized Drazin-Riesz inverse of AC then T := BS 2 D is a generalized Drazin-Riesz inverse of BD.
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