Alternative Characterization of Polynomial Ideal

Theorem

Let Polynomial Ideal $M\in \mathbb{F}[x]$. The following are equivalent:

1. $M$ is an Ideal
2. $M$ is non-empty subset such that:
   1. $f\in \mathbb{F}[x]$, $g\in M$, then $fg=gf\in M$
   2. $f,g\in M$, then $f+g\in M$ Very similar to the definition of a Subspace