I am trying to understand the proof of the Cayley-Hamilton Theorem given in Section 4 of this document.
On pg. 4 of the document, there is a line which reads:
From general multilinear algebra, we know that an action of $R$ on $M$ induces an adjugate action on $M$ as well. For each $r\in R$, the adjugate $r^{adj}$ need not lie in the subring $\text{End}_{k[x]}M$ of $R$, nor need it commute with $R$.
I do not know anything about the adjugate action the author has mentioned. Can somebody please point towards an appropriate reference to gain relevant knowledge.
Thanks.