  Let R be a local complete ring. For an R-module M the canonical ring map R\to
End_R(M) is in general neither injective nor surjective; we show that it is
bijective for every local cohomology module M := H^h_I(R) if H^l_I(R) = 0 for
every l\neq h(:= height(I)) (I an ideal of R); furthermore the same holds for
the Matlis dual of such a module. As an application we prove new criteria for
an ideal to be a set-theoretic complete intersection.
