# Cayley-Hamilton over super rings

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?

The case when $$R$$ is purely even but the module has a $$Z/2$$-grading was studied before, see for example