**Jategaonkar’s lemma**. Let be a ring and a subring of If are in the centralizer of in and are left or right linearly independent over then

*Proof.* I will assume that are right* *linearly independent over The argument for the “left” version is identical. We basically need to prove that the set of all monomials in is an -basis for the ring generated by over So suppose that the claim is false. Then there exists a non-zero of minimum total degree such that Write

where and with Now

Thus because are right linearly independent over Again, we can write

where and Then

and so, since are right linearly independent over we get But that contradicts the minimality of the total degree of because

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