The example is given in the following picture:
Here is the text for clarity
EXAMPLES. If $I$ is a left ideal of a ring $R$, then $I$ is a left $R$-module with $ra(r \varepsilon R,a \varepsilon I)$ being the ordinary product in $R$. In particular, $0$ and $R$ are $R$-modules. Furthermore since $I$ is an additive subgroup of $R$, $r / I$ is an (abelian) group. $R/I$ is an $R$-module with $r(r_1+I) = rr_1 + I$. $R/I$ need not be a ring however unless $I$ is a two sided ideal.
It is not clear for me how "$I$ is a left $R$-module with ra being the ordinary product in $R$", I know that by the definition of an ideal it is a ring and hence an additive abelian group but how the ring action is applied it is not clear for me the details, how multiplication of a ring element is distributed over addition of 2 ideal elements and so on, could anyone clarify this for me please?
Also,I did not understand why "$R/I$ need not be a ring, however, unless $I$ is a two-sided ideal" could anyone explain this for me please?