Let $S$ be a subset of the ring $R$; we say that $S$ is multiplicative if
(a) $0 \notin S$,
(b) $1 \in S$, and
(c) whenever $a,b\in S$, we have $ab \in S$.
We can merge (b) and (c) by stating that $S$ is closed under multiplication, if we regard $1$ as the empty product.
Here are some standard examples of multiplicative sets.
- The set of all nonzero elements of an integral domain.
- The set of all nonzero elements of a commutative ring $R$ that are not zero divisors.
- $R \setminus P$, where $P$ is a prime ideal of the commutative ring $R$.
I think 3 is not a correct example as $R \setminus P$ is an integral domain and has zero element $P$. It should be $R \setminus P$ except $P$. Is it right?