Let $R$ be a ring and if $R= \bigoplus R_i$ as additive subgroups for each $i \in I$ where $I $ is a finite group
Is that implies $R_i R_j $ must be contained in $R_p$ for any $p \in I$ ??
In another word, if $r_1 \in R_i$ and $r_2 \in R_j$ is that implies
$r_1 r_2 = r \in R_p $??, where the multiplication between $R_i ,R_j $ is the multiplication defined on $R$
Please I need any hint
I think your question is: if $R=\bigoplus_{i} R_i$, for $i\in I$ where $I$ is a finite group, is it true that for every $i$ and $j$, $R_iR_j\subseteq R_k$ for some $k\in I$? This is sort of an odd question, because it doesn't matter whether $I$ is a group or not in this question, the question is the same if you allow $I$ to be a finite set. The group structure is irrelevant.
Let $R=\mathbb{Q}^2$ with the product ring structure. Choose any two linearly independent vectors $v$ and $w$. Then $\newcommand{\QQ}{\mathbb{Q}}\QQ^2 = \QQ v\oplus \QQ w$, so your question becomes is it always true that $vw$ is a scalar multiple of $v$ or $w$. The answer of course is no. Let $v=(1,2)$, $w=(2,1)$. Then $vw=(2,2)$, which is not a scalar multiple of either.
Anyway, if you can reformulate your question a little more coherently, I might be able to be more useful. All I can do is answer what's asked right now, as best as I can understand it.