\begin{definition}[Definition:Multiplicatively Closed Subset of Ring]
Let $\left({A, +, \circ}\right)$ be a ring with unity $1_A$ and zero $0_A$.
Let $S \subseteq A$ be a subset.
Then $S$ is '''multiplicatively closed''' {{iff}}:
:$(1): \quad 1_A \in S$
:$(2): \quad x, y \in S \implies x \circ y \in S$
\end{definition}