On the Product of Congruence Classes over $\mathbb{Z}$

Question

Is it possible to multiply an element $a$ of $\mathbb{Z}_4$ to an element $b$ of $\mathbb{Z}_2$? If so, what are the needed conditions? To which set ($\mathbb{Z}_4$ and $\mathbb{Z}_2$) does $a\cdot b$ belong?

Answer 1

Yes, it is possible. $ab$ will belong to $\mathbf{Z}_2$ in this case. This can be done with any elements $b \in \mathbf{Z}_2$ and $a \in \mathbf{Z}_4$. The reason this is possible is because $2$ divides $4$.

However, before multiplying an element of $\mathbf{Z}_2$ by an element of $\mathbf{Z}_4$, a mathematician would probably say something like "Consider the $\mathbf{Z}_4$-module structure on $\mathbf{Z}_2$," because it's not that common.