How can $\Bbb{Z}/10$ be viewed as a $\Bbb{Z}$-module?

Question

How can $\Bbb{Z}/10$ be viewed as a $\Bbb{Z}$-module? For example, when I compute $5.\overline{5}$, where $5\in\Bbb{Z}$ and $\overline{5}\in\Bbb{Z}/10$, is this equal to $5$?

Answer 1

Exactly ! Z acts on Z/5Z by multiplication.

Answer 2

Every abelian group $A$ is a $\mathbb Z$-module in a unique way, defining the multiplication as $$ \mathbb Z\times A \to A,\, (k,a) \mapsto ka := \underset{k \ \text{factors}} {\underbrace{a+\dots+a}}. $$ So in your case $5 \cdot \bar 5 = \bar 5 + \bar 5 + \bar 5 + \bar 5 + \bar 5 = \bar 5$.

Actually, $\mathbb Z/10\mathbb Z$ is a ring, and as such is a $\mathbb Z$-algebra (not just module) in a unique way.