Remark

Let $K'$ be a commutative K-algebra. Every element $t$ of $A_{(K')}$ can be written as $\sum_{i\in I}t_i\otimes e_i$, where $(t_i)\in {K'}^I$. It follows from the proof of Lemma 5 that the characteristic polynomial Pc$_{A_{(K')}/K'}(t; X)$ is equal to $P((t_i),X)^n$.