According to the definition of Noetherian ring, a ring is said to be Noetherian if all increasing chains of ideals eventually stabilize.
Now, let $R$ be a commutative ring with identity and we take a collection $C$ of proper ideals of $R$. Consider an increasing chain of ideals $$I_1\subseteq I_2\subseteq I_3\subseteq ....$$ Now, $I=\cup_{i}I_i$ is a maximal ideal, as any proper ideal containing $I$, is in $C$ and falls in the increasing chain, hence it falls inside $I$ and if, $I=R$, then $1\in I$, therefore, $1\in I_i$ for any $i$, which again contradicts the definition of $I_i$.
So, does all commutative rings with identity falls inside Noetherian rings?