Quotient ring, example about $( r_1 +I)( r_2 +I)\ne r_1r_2+I$

Question

Could one construct a ring $R$ and an two-sided ideal $I$ in $R$ such that there exist $r_1, r_2\in R$ with $(r_1 +I)(r_2 +I)\ne r_1r_2+I$? (Here $(r_1 +I)(r_2 +I)=\{( r_1 +a)( r_2 +b)\mid a,b\in I\}$.)

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Well, I am learning about Quotient ring. We know that $(r_1 +I)(r_2 +I)\subseteq r_1r_2+I$.

Answer 1

Take $R=\mathbb{Z}$, $I=2\mathbb{Z}$ and $r_1=r_2=0$. Note that $I^2=4\mathbb{Z}\subsetneq I$.