Example of ordinals $a,b,c$ such that $(a + b) \cdot c \neq a\cdot c + b\cdot c$ I know that $c\cdot(a +b)=c\cdot a+c\cdot b$, but I don't see the counterexample to the hypothesized property in the question.

By definition, we have that $(\omega_0+1)\cdot\omega_0$ is the supremum of $\{(\omega_0+1)\cdot n\mid n\text{ is finite}\}$. This supremum is $\omega_0^2$, not $\omega_0^2+\omega_0$.
