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Let H be a Hopf algebra over a field k such that every finite-dimensional (left) H-module is semisimple. We give a counterpart of the first fundamental theorem of the classical invariant theory for locally finite, finitely generated (commutative) H-module algebras, and for local, complete H-module algebras. Also, we prove that if H acts on the k-algebra A = k[[X 1 , . . . , X n ]] in such a way that the unique maximal ideal in A is invariant, then the algebra of invariants A H is a noetheriandoi:10.4064/cm88-1-5 fatcat:w3lvbzqjkzc2vepggvxkn7pjve