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?



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


["On the Cayley-Hamilton equation in the supercase" by Issai Kantor and Ivan Trishin, Comm. in Algebra 27:1 (1999), 233-259](https://www.tandfonline.com/doi/abs/10.1080/00927879908826430)
["Berezinians, Exterior Powers and Recurrent Sequences", by
H. M. Khudaverdian and TH. TH. Voronov,
Letters in Mathematical Physics, 74 (2005), 201–228](https://link.springer.com/article/10.1007/s11005-005-0025-7)


Already in this case it seems that you should not expect the polynomial to be monic.


