For each proof relating to this there always seems to be a part where they assume without loss of generality that the ideal they are trying to find the primary decomposition of is the zero ideal. For example in my courses notes they first proof a lemma saying:
If $R$ is a Noetherian ring and $I$ is a proper ideal of $R$, suppose $I$ is not primary then there exists non-equal ideals $J,K$ such that $J\cap K = I$.
The first line of the proof states, 'We know the quotient ring of a Noetherian ring is Noetherian. And we know there is a bijection between the ideals of $R/I$ and the ideals of $R$ containing $I$ hence we may assume w.l.o.g that $I=(0)$.'
I do not see why this can be assumed.
If you can do it for the zero ideal of any noetherian ring, then you can do it for any ideal in a noetherian ring. For suppose you have the noetherian ring $R$ and the nonzero ideal $I$. Let $S=R/I$; then $S$ is noetherian. Find the primary decomposition of the zero ideal in $S$. Then use the lattice isomorphism theorem to “lift” the primary decomposition of the zero ideal of $S$ to a bunch of ideals of $R$ that contain $I$. They will give you the primary decomposition of $I$, thanks to the Lattice Isomorphism Theorem.