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Two fractions $r_1/s_1$ and $r_2/s_2$ in the localization $S^{-1}R$ are defined to be equivalent if there exists an element $t$ in the multiplicatively closed set $S$ such that $t(r_1s_2 - s_1r_2) = 0$. This relation ensures that the fractions represent the same element in the localization.

Conditions: $R$ is a commutative ring.; $S \subset R$ is a multiplicatively closed subset.; $r_1, r_2 \in R$ and $s_1, s_2 \in S$.

Two fractions $r_1/s_1$ and $r_2/s_2$ in the localization $S^{-1}R$ are defined to be equivalent if there exists an element $t$ in the multiplicatively closed set $S$ such that $t(r_1s_2 - s_1r_2) = 0$. This relation ensures that the fractions represent the same element in the localization.

Conditions: $R$ is a commutative ring.; $S \subset R$ is a multiplicatively closed subset.; $r_1, r_2 \in R$ and $s_1, s_2 \in S$.

The naive condition $r_1s_2 = s_1r_2$ is insufficient because in a general commutative ring, elements can be annihilated by nonzero elements of $S$. The refined condition requires the existence of some $t \in S$ such that $t(r_1s_2 - s_1r_2) = 0$, which accounts for zero-divisor phenomena and ensures the equivalence relation is well-defined.

Conditions: $R$ is a general commutative ring (not necessarily an integral domain).; $S \subset R$ is multiplicatively closed.

The naive condition $r_1s_2 = s_1r_2$ is insufficient because in a general commutative ring, elements can be annihilated by nonzero elements of $S$. The refined condition requires the existence of some $t \in S$ such that $t(r_1s_2 - s_1r_2) = 0$, which accounts for zero-divisor phenomena and ensures the equivalence relation is well-defined.

Conditions: $R$ is a general commutative ring (not necessarily an integral domain).; $S \subset R$ is multiplicatively closed.

Two fractions $r_1/s_1$ and $r_2/s_2$ are equal in the localization $S^{-1}R$ exactly when there exists an element $t \in S$ such that $t(r_1s_2 - s_1r_2) = 0$. This refines the naive cross-multiplication rule to handle zero-divisor phenomena in general commutative rings.

Conditions: $R$ is a commutative ring.; $S \subset R$ is multiplicatively closed.; $r_1, r_2 \in R$ and $s_1, s_2 \in S$.

Two fractions $r_1/s_1$ and $r_2/s_2$ are equal in the localization $S^{-1}R$ exactly when there exists an element $t \in S$ such that $t(r_1s_2 - s_1r_2) = 0$. This refines the naive cross-multiplication rule to handle zero-divisor phenomena in general commutative rings.

Conditions: $R$ is a commutative ring.; $S \subset R$ is multiplicatively closed.; $r_1, r_2 \in R$ and $s_1, s_2 \in S$.