ideal of annihilating polynomials
Gather every polynomial that annihilates a fixed operator T into one collection, written I(T) = { p in F[x] : p(T) = 0 }. The previous term observed that this collection is an ideal of the ring F[x]; this term names it, studies its shape, and extracts the punchline. Think of it as the operator's algebraic signature, recorded as a set of polynomials.
F[x], the polynomials in one variable over a field F, is a principal ideal domain (PID): every ideal is the set of multiples of one single generator. So I(T) = (g) for some polynomial g — and because nonzero ideals have a unique monic generator, g is forced to be the minimal polynomial m_T. The abstract algebra of ideals collapses to one concrete object you can compute.
What makes F[x] a PID is the division algorithm: given any p and any nonzero d, you can write p = qd + r with deg r < deg d. Run this with d = m_T: the remainder r also annihilates T (since p and qm both do), but r has degree below the minimal degree, forcing r = 0. That is exactly why m divides every annihilator — the PID structure is the engine.
Conceptually this is where linear algebra hands the wheel to ring theory. The single ideal I(T) encodes the operator's annihilation behavior, the quotient ring F[x]/I(T) is isomorphic to the algebra F[T] of polynomials in T, and these ideas scale up: replacing F[x] by a general PID gives the structure theorem behind rational canonical form.
The annihilator ideal is exactly the multiples of the minimal polynomial, and quotienting F[x] by it reproduces the algebra of all polynomials in T.
I(T) is never the whole ring (the constant 1 does not annihilate a nonzero space) and never just {0} in finite dimension (Cayley-Hamilton supplies a nonzero member). So it is a genuine proper nonzero ideal — exactly the setting where the monic generator is unique.