If $R$ is a commutative ring, then the Cayley-Hamilton theorem states that any endomorphism $\phi: R^{n} \rightarrow R^{n}$ of a rank $n$ free module satisfies its own characteristic polynomial, in particular satisfies a monic degree $n$ polynomial.

Now suppose that instead $R = R_{0} \oplus R_{1}$ is a commutative super ring (that is, it is $\mathbb{Z}/2$-graded, with even degree elements in the center, and odd degree elements anticommuting with each other).

Does every endomorphism $\phi: R^{p, q} \rightarrow R^{p, q}$ of a free module on $p$ even and $q$ odd variables satisfy a monic polynomial over $R$ of degree $p+q$? Does it satisfy *some* monic polynomial?