if $\cdot$ and $\odot$ are associative operations on $\mathbb{Z}$ when is the sum $(\cdot + \odot)$ associative?

Question

Where $a(\cdot + \odot)b$ is defined as $(a\cdot b) + (a\odot b)$.

I know if $\cdot$ and $\odot$ distribute through addition (i.e. $a\cdot(b+c)=a\cdot b+ a\cdot c$) then the sum $(\cdot + \odot)$ is associative, but this isn't a necessary condition, since if we define $\cdot$ to be addition and $\odot$ to be multiplication the operation $a(\cdot + \odot)b=(a+b)+(ab)$ is associative but addition does not distrubt through addition (i.e. a+(b+c)$\neq$(a+b)+(a+c)) however multiplication does.

Is it possible that (a+b)+(ab) can be rewritten as the sum of operations which distribute over addition? IF so can distributing over addition be a necessary and sufficient condition for the sum of operations to be associative. If this is the case could someone calculate and explain, otherwise what is a neccisary condition(s) on $\cdot$ and condition(s) on $\odot$ which together make $\cdot + \odot$ associative?

Answer 1

Use the associative properties you have of the three operations involved: $$ \begin{array}{rclcr} \alpha + ( \beta +\gamma ) &=& (\alpha +\beta)+\gamma & \quad \mbox{ for all } \quad &\alpha,\beta,\gamma\in\mathbb{Z} \\ \\ i \bullet ( j \bullet k ) &=& (i \bullet j)\bullet k & \quad \mbox{ for all } \quad &i,j,j\in\mathbb{Z} \\ \\ p \odot ( q \odot r ) &=& (p \odot q)\odot r & \quad \mbox{ for all } \quad &p,q,r\in\mathbb{Z} \\ \end{array} $$

And now? What equality involving three integers $a$, $b$ and $c$ characterizes the operation $(\bullet + \odot ) $ as an associative operation? Write this equality. Then express both members of this equality and in terms of operations $\bullet$, $+$ and $\odot$.

Spoiler:

$$ a (\bullet + \odot ) \Big( b (\bullet + \odot ) c \Big) = \Big(a (\bullet + \odot ) b \Big)(\bullet + \odot ) c \quad \mbox{ for all } \quad a,b,c\in\mathbb{Z} $$