Reduced Primary Decomposition in Almost Clean Rings

Question

Theorem 2.3. For a commutative ring $R$ the following conditions are equivalent.

1. $R$ is an indecomposable almost clean ring.
2. For $x\in R$, either $x$ or $x−1$ is regular.
3. For (prime) ideals $I$ and $J$ of $R$ consisting of zero divisors,$I+J=R$.

Corollary 4. Suppose that $R$ has a reduced primary decomposition $0=Q_1∩···∩Q_n$ where $Q_i$ is $P_i$-primary. Then $R$ is an indecomposable almost clean ring if and only if $Q_i+Q_j\ne R$ (or equivalently, $P_i+P_j\ne R$).

Found this on Weakly Clean Rings and Almost Clean Rings by Ahn & Anderson.

I believe that proving Corollary 4 is using Theorem 2.3 (1) and (3), i.e. $P_i$ and $P_j$ must consist of zero divisors. But I dont have any idea about how to obtain these conditions.
Question (edited):
Is if $P_i$ is a prime ideal then $P_i⊆Z(R)$? Or is it because $0=Q_1∩ ... ∩Q_n$ then $P_i⊆Z(R)$. Any explanation about how prove Corollary 4?