On endomorphism rings of local cohomology modules

Let $R$ be a local complete ring. For an $R$-module $M$ the canonical ring map $R\to \mathrm{End}_R(M)$ 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 $l\neq h$ $(=\mathrm{height} (I))$ ($I$ 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.