Image of Jacobson Radical is the Jacobson Radical

You have to show $$\mathfrak m_1\cap\cdots\cap\mathfrak m_s+\mathfrak a=\mathfrak m_1\cap\cdots\cap\mathfrak m_t,$$ where $s\ge t$, $\mathfrak a\subseteq\mathfrak m_i$ for $1\le i\le t$, and $\mathfrak a\nsubseteq\mathfrak m_j$ for $t+1\le j\le s$.

The inclusion "$\subseteq$" is obvious.

For the converse notice that $\mathfrak m_{t+1}\cap\cdots\cap\mathfrak m_s+\mathfrak a=R$ hence $1=a+b$ with $a\in\mathfrak a$ and $b\in\mathfrak m_{t+1}\cap\cdots\cap\mathfrak m_s$. Let $x\in m_1\cap\cdots\cap m_t$. Then $x=ax+bx$ with $ax\in\mathfrak a$ and $bx\in\mathfrak m_1\cap\cdots\cap\mathfrak m_s$.