Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\mathop{\mathrm{det}}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.

I am looking for a proof that: $-\mathop{\mathrm{adj}}(A) = p_1 I + p_2 A + \dots + p_n A^{n-1}$.
In the case where $\mathop{\mathrm{det}}(A)$ is a unit, $A$ is invertible, and the proof follows from the Cayley-Hamilton theorem. But what about the case where $A$ is not invertible?

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.

I am looking for a proof that $-\operatorname{adj}(A) = p_1 I + p_2 A + \dots + p_n A^{n-1}$.

In the case where $\operatorname{det}(A)$ is a unit, $A$ is invertible, and the proof follows from the Cayley–Hamilton theorem. But what about the case where $A$ is not invertible?