Definition:Multiplicatively Closed Subset of Ring
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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 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.
Also see
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.
Sources
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