Definition:Multiplicatively Closed Subset of Ring

Definition
Let $\struct {A, +, \circ}$ be a ring with unity $1_A$ and zero $0_A$.

Let $S \subseteq A$ be a subset.

Then $S$ is multiplicatively closed :


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

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.

Also see

 * Definition:Localization of Ring
 * Definition:Saturation of Multiplicatively Closed Subset of Ring

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

Consider the ring of integers $\struct {\Z, +, \times}$.

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

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

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