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Let $R$ be a ring, $A$ a (not necessarily commutative) $R$-algebra and $M$ a (left) A-module which is free of finite rank as an $R$-module. If $a\in A$ then multiplication with $a$ on $M$ is an $R$-linear endomorphism of $M$ and as such it has a characteristic polynomial $\chi_a$. I've learned from notes by Bart de Smit which I've accidentally found via Google (http://www.math.leidenuniv.nl/~desmit/notes/charpols.pdf) that the function $\chi_\bullet$ is determined by its values on a generating set of $A$ as an $R$-module. He proves this by deriving suitable formulas for the characteristic polynomial of a sum of two endomorphisms.

I'm looking for a reference for this fact that can be cited more easily than the informal notes above.

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up vote 4 down vote accepted

One reference is: S. A. Amitsur, On the Characteristic Polynomial of a Sum of Matrices, Linear and Mult. Algebra 8 (1980), 177-182. (pp. 469-474 in Selected Papers of S. A. Amitsur, Part 2, AMS 2001.)

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Great, thank you very much. –  Philipp Hartwig Jul 15 '11 at 15:35
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