Let $R$ be a commutative ring with unit element. A nonempty subset $S$ of $R$ is called multiplicative system if
$1. \ 0\notin S$
$2. \ s_1,s_2\in S$ implies that $s_1s_2\in S$.
Let $n=pq$ where $p,q$ are prime numbers. We know that $\mathbb{Z}_n$ is a ring with addition and multiplication modulo $n$. Consider the $\mathbb{Z}_n^{\times}=\{x: 1\leq x<n, \ \text{gcd}(x,n)=1\}$.
It's easy to check that $\mathbb{Z}_n^{\times}$ is multiplicative system.
What if we consider $\mathbb{Z}_n^{\times}\cup \{p\}$? I have checked that this is not multiplicative system.
Am I right? If yes, can anyone suggest some other example of multiplicative system in $\mathbb{Z}_n$ distinct from $\mathbb{Z}_n^{\times}$?
The multiplicative closure of $\{p\}\cup \Bbb{Z}_n^\times$ , namely $\{x: 1\le x< n, \gcd(x,q)=1\}$ is a multiplicative system distinct from $\Bbb{Z}_n^\times$.