In most Linear Algebra-courses, the theory of characteristic polynomials of endomorphisms is only treated in the case of $F$-vector spaces $V$, where $F$ is a field. However, in more advanced books, results are used in more general context, where $R$ is a (specific) ring and $M$ is an $R$-module. I am looking for a comprehensive book that covers the theory in a more general context.
So far, I checked the following books: Adkins-Weintraub: Algebra; Bosch: Lineare Algebra; Bourbaki: Algebra I-III; Hoffmann-Kunze: Linear Algebra; Lang: (Linear) Algebra. However, they treat most results only over fields.
To be more specific: I was wondering, whether the following the following two propositions hold in a more general setting:
Let $A: V\to V$ be an endomorphism of a finite-dimensional $F$-vector space $V$. Suppose there is a monic polynomial $p\in F[T]$, such that $V$ is isomorphic to $F[T]/(p)$ as a $F[T]$-module (via A). Then $p$ is both the minimal polynomial and the characteristic polynomial of $A$.
from Bosch's Lineare Algebra and
Let $V=V_1\oplus\ldots\oplus V_r$ be a a direct sum of finite dimensional $F$-vector spaces with an endomorphism $A:V\to V$. Assume $AV_i\subset V_i$ and let $\chi_{A_i}(T)$ the characteristic polynomial of $A$ on $V_i$. Then $$ \chi_A(T)=\prod_i \chi_{A_i}(T),$$ thus the characteristic polynomial is multiplicative under direct sums.
from Lang's Algebra.