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A subset $S$ of a ring $R$ is multiplicatively closed if it is stable under multiplication and contains the multiplicative identity. Specifically, if two elements are taken from $S$, their product also sits in $S$, and the identity element belongs to $S$.

Conditions: $S$ is a subset of the commutative ring $R$.; Products of elements of $S$ remain in $S$.; The identity element $1$ belongs to $S$.

A subset $S$ of a ring $R$ is multiplicatively closed if it is stable under multiplication and contains the multiplicative identity. Specifically, if two elements are taken from $S$, their product also sits in $S$, and the identity element belongs to $S$.

Conditions: $S$ is a subset of the commutative ring $R$.; Products of elements of $S$ remain in $S$.; The identity element $1$ belongs to $S$.

The localization $S^{-1}R$ is defined as the collection of formal fractions $r/s$ where the numerator $r$ is an element of the commutative ring $R$ and the denominator $s$ is an element of the multiplicatively closed subset $S$.

Conditions: $R$ is a commutative ring.; $S \subset R$ is a multiplicatively closed subset.; $r \in R$ and $s \in S$.

The localization $S^{-1}R$ is defined as the collection of formal fractions $r/s$ where the numerator $r$ is an element of the commutative ring $R$ and the denominator $s$ is an element of the multiplicatively closed subset $S$.

Conditions: $R$ is a commutative ring.; $S \subset R$ is a multiplicatively closed subset.; $r \in R$ and $s \in S$.