Let $R$ be a unital ring and denote its center by $Z(R)$. If $I$ is an ideal of $Z(R)$, then the set $RI$ (consisting of finite sums of elements of the form ra where $r\in R$ and $a\in I$) is clearly an ideal of $R$.
My question is the following:
If $I$ is a proper ideal of $Z(R)$, is $RI$ necessarily a proper ideal of $R$?
The proof that I had in mind does not seem to work out, and I am now suspecting that the answer is negative. Are there any nice and intuitive counter-examples?