Let $n$ be a positive integer and $k \in \Bbb Z$. Show that $\langle [k]_n \rangle$ is an ideal for $(\Bbb Z_n, +, \cdot)$.
I'm always confused when we have two binary operations. How am I to know if by $\langle [k]_n \rangle$ they mean $$\langle [k]_n \rangle=\{[k]_n^t \mid t \in \Bbb Z \}$$ or $$\langle [k]_n \rangle = \{t[k]_n \mid t \in \Bbb Z\}?$$ These are both well defined so either one of them work. If I suppose it's the latter one then, for $a,b \in \langle [k]_n \rangle$ I have that $a= t[k]_n$ and $b=s[k_n]$ for $t,s \in \Bbb Z$. Now $$a+(-b)=t[k]_n - s[k]_n = [tk-sk]_n=[(t-s)k]_n \in \langle [k]_n \rangle$$ as $(t-s)$ is an integer. Now for $x \in \langle [k]_n \rangle$ and $y \in \Bbb Z_n $ I have that $xy=t[k]_n \cdot [m]_n = [tkm]_n \in \langle [k]_n \rangle$ and $yx = [mtk]_n \in \langle [k]_n \rangle$. But how am I to know what operation should I consider with the cyclic group?