Tensor product of local Artinian rings

Question

Consider a complete Noetherian local ring $R$ and two local Artinian $R$-Algebras $A$ and $B$. I'm trying to prove that the spectrum $\text{Spec}(A\otimes_{R}B)$ is connected or, equivalently, that $A\otimes_{R}B$ has only trivial idempotents. I suppose we can base change to the residue field $k$ of $R$, prove the same result for local Artinian $k$-Algebras, and then come back via Hensel's Lemma. But I don't have any idea for proving that also in the case of $k$-Algebras. Could you help me please? Thanks!