You are right. I added the global dimension condition in the final step and I didn't notice that this ring is a regular local ring and o it is CM. But can we give an example of an algebra as above with infinite global dimension? I mean, I am looking for an algebra with the above properties with infinite global dimension over a complete regular local ring.

Thank you so much. So if we add the condition that $R$ is a semisimple ring we can prove the claim. Is it true?

Thank you so much. Can you see the following question?mathoverflow.net/questions/277378/…

Thank you very much. But how can I prove this?

$\Lambda$ is finitely genreated as an $R$-module. In fact, $\Lambda$ is a noether $R$-algebra.