Definition:Multiplicatively Closed

Definition

Let $A$ be a ring with unity $1$.

Let $S \subseteq A$ be a subset.

We say that $S$ is multiplicatively closed (often abbreviated to m.c.) if:

• $1 \in S$
• If $x,y\in S$ then $xy \in S$

Some texts additionally require that $0 \notin S$.

Also see

Compare with closed in the general context of abstract algebra. The difference is subtle.

Consider the ring of integers $\left({\Z, +, \times}\right)$.

For $n \ne 1$, consider the set of integer multiples $n \Z$.

Then $1 \notin n \Z$, but $\left({n \Z, \times}\right)$ is closed from Integer Multiples Closed under Multiplication.

So $n \Z$ is closed under $\times$ but not actually multiplicatively closed as such.