Operations at ideals.
The sum is defined as $$I_1 + I_2 + \dots + I_m =\{a_1+a_2+\cdots +a_m\mid a_i \in I_i\}.$$ It can be proven that $$I_1 + I_2 + \dots + I_m \trianglelefteq R$$ and each $$I_i\subseteq I_1 + I_2 + \dots + I_m. $$
The product is defined as $$I_1 \cdot I_2 \cdot \dots \cdot I_m =\{\text{finite sum of products } a_1\cdot a_2 \cdot \dots \cdot a_m \mid a_i \in I_i\}.$$ It can be proven that $$I_1 \cdot I_2 \cdot \dots \cdot I_m \trianglelefteq R$$ $$I_1 \cdot I_2 \cdot \dots \cdot I_m \subset I_i.$$
Could you explain to me why the subsets hold??
If $a_i\in I_i$, $a_i=0+\cdots+a_i+\cdots+0\in I_1+\cdots+I_m$.
If $a_i\in I_i$ for each $1\le i\le m$, then $a_1\cdot\ldots\cdot a_m\in I_1\cap\cdots\cap I_m$.