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Localization is considered a generalization because it introduces fraction-like elements into a general commutative ring, whereas the field of fractions construction only applies to integral domains. When $R$ is an integral domain and $S$ is chosen to be the set of nonzero elements, the localization $S^{-1}R$ recovers the familiar field of fractions.

Conditions: $R$ is a commutative ring.; In the special case, $R$ is an integral domain and $S$ is the set of nonzero elements of $R$.

Localization is considered a generalization because it introduces fraction-like elements into a general commutative ring, whereas the field of fractions construction only applies to integral domains. When $R$ is an integral domain and $S$ is chosen to be the set of nonzero elements, the localization $S^{-1}R$ recovers the familiar field of fractions.

Conditions: $R$ is a commutative ring.; In the special case, $R$ is an integral domain and $S$ is the set of nonzero elements of $R$.

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 $s \in S$.