For a (left) ideal $I$, how can $R \cdot I$ be a proper subset? I have the following definition for a left ideal: An additive subgroup $I$ of a ring $R$ is called left ideal, if $R \cdot I \subseteq I$ where $R \cdot I = \{ r \cdot i \,\,\,|\,\,\, r \in R, i \in I\}$.
My question is: how can $R \cdot I$ be a proper subset of $I$? My first observation is that if this is indeed the case, $R$ isn't allowed to have a neutral element for multiplication, because else $R \cdot I \supseteq 1 \cdot I = I \Rightarrow R=I$. 
Is there an easy example?

Take $R = 2\mathbb Z, I = 8 \mathbb Z.$ Then $8 \notin R \cdot I.$
