Module-vs-vector space, a sentence in the book by Blyth

Question

I am now reading the book by Blyth, 'Module thery---an approach to linear algebra'.

I do not understand such a paragraph with which he motivates the learning of module theory:

In many ways, however, the notion of a vector space is too restrictive. Perhaps one way we can illustrate this is by asking the following question: has it occurred to the reader that, in most elementary introductions to linear algebra, determinants are defined for square matrices whose entries belong to a given field yet, come the consideration of eigenvalues, the matrix whose determinant has to be found has its entries in a polynomial ring? Put another way, should the various properties of determinants not really be developed in a more general setting?

Can anyone explain it in detail?

Answer 1

The eigenvalues of a matrix $A$ over a field $k$ are the roots of the characteristic polynomial $\chi(A)=\det(\lambda I-A)$. But what is this thing we're taking the determinant of? $\lambda$ is not an element of $k$, but an indeterminate, so to properly describe $\lambda I-A$ we ought to say its coefficients, e.g. $2-\lambda$, are elements of the polynomial ring $k[\lambda]$. So this is an example of a determinant of a matrix with entries from a ring that's not a field.