I'm currently going through the section in Dummit and Foote regarding the structure of commutative artinian rings with unity. It has already shown that there exists finitely many distinct maximal ideals $M_1,\ldots,M_n$. It then defines $Jac(R)=M_1 \cap M_2 \cap \cdots \cap M_n$, and the following statement is given:
$$ \prod_{i=1}^n M_i^m \subseteq \left( \prod_{i=1}^n M_i \right)^m \subseteq Jac(R)^m$$
My question is, why does the book say "$\subseteq$" and not "="? Aren't these ideals all equal? The first equality follows from the fact that ideal multiplication is commutative, and the second follows from the fact that any two distinct maximal ideals must be coprime, so $M_1M_2\cdots M_n=M_1\cap M_2\cap\cdots\cap M_n$.