# Definition:Multiplicatively Closed Subset of Ring

## Definition

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 if and only if:

$(1): \quad 1_A \in S$
$(2): \quad x, y \in S \implies x \circ y \in S$

## Also defined as

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

## Also known as

The term multiplicatively closed is often abbreviated to m.c.

## Example

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.