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For every ring R, it can be shown that $ a cdot 0 = 0 $ for all $ a in R $, making use of the distributive law.

In a near-ring R, this does not necessarily hold, as the following example shows:
Let $G$ be a group with at least two elements, and let $ F := {fcolon G to G} $ be the set of all functions from $G$ to $G$. Together with the usual addition on functions, i.e. for $ f, g in F $ the sum is defined as $ f + gcolon G to G, a mapsto f(a) + g(a) $, and the composition as the multiplication, F is a near-ring. As the left distributive law does not hold on $F$, $ f circ 0 = 0 $ does not necessarily hold, with $ 0 in F $ being the constant mapping to $ 0 in G $. This can be seen by choosing $f$ as a constant function which does not map to $0$: we have $ f circ 0 = f neq 0 $. However, $ 0 circ 0 = 0 $ still holds. The property $ 0 cdot 0 = 0 $ can indeed be shown to hold for near-rings in general.

Now, I wonder if there is a (more or less commonly-studied) ring-like structure in which $ 0 cdot 0 = 0 $ does not hold. Do you know any such example?