nLab multiplicative subset

Definition

For $R$ a commutative ring, a subset $S \hookrightarrow R$ of its set of elements is called multiplicative if it is closed under multiplication “$\cdot$”in $R$, hence if $s_1,s_2 \in S \hookrightarrow R$ implies that $s_1 \cdot s_2 \in S \hookrightarrow R$, or equivalently, that $S$ forms a subsemigroup of $R$. Usually one also demands that the unit element is in the set $S$, or equivalently, that $S$ forms a submonoid of $R$.

Every two-sided filter of a ring in a commutative ring is a multiplicative subset.

The localization of a commutative ring at a multiplicative subset exists, denoted $S^{-1}R$ or $R[S^{-1}]$.

See also

• filter of a ring
• multiplicative subset of cancellative elements

References

• The Stacks Project, def. 10.9.1