Submodules/invariant subspaces

Given a module, it's natural to study its submodules.
For $K[T]$, that corresponds to:
- $T$-invariant ($T$ preserves 'em)
- subspaces ($K$ preserves 'em)

This is -exactly- the natural thing to study from the linear algebra
POV (i.e., down and dirty and concrete).

Most naively, you have
$\ker T < \dots < \ker T^k < \dots \ker T^n$
$\im T > \dots > \im T^k > \dots \im T^n$
(BTW, $\ker T^n \oplus \im T^n = V$,
 and under $\ker T^n = V/\im T^n$, these are the same flag,
 if you have the right indexing -- so the only data is:
 - the direct sum $\ker \oplus \im$
 - the flag on $\ker$)

More generally, you have
$\ker (T - \lambda I)$ are the eigenspaces
$\ker (T - \lambda I)^n$ are the generalized eigenspaces
...and in between you have a whole flag of kernels.

...and this all yields the generalized eigenspace decomposition etc.,
a la Axler.
(who ignores all the intermediary flags)

   %   %   %   %   %   %   %   %   %   %   %   %   %   %   %
