What is the general criterion for equality of two fractions $r_1/s_1$ and $r_2/s_2$ in a localization?

Conditions

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

Reasoning, step by step

1. Start with two arbitrary fractions $r_1/s_1$ and $r_2/s_2$.
2. Compute the cross-product difference $r_1s_2 - s_1r_2$.
3. Check if there exists an element $t \in S$ that annihilates this difference, i.e., $t(r_1s_2 - s_1r_2) = 0$.
4. Conclude that the fractions are equivalent if and only if such a $t$ exists.

Example

The board writes $\frac{r_1}{s_1} = \frac{r_2}{s_2}$, then $(r_1 s_2 - s_1 r_2)=0$, then $\exists t \in S : t(r_1 s_2 - s_1 r_2) = 0$. The speaker explains that equality holds not only when the cross product itself vanishes, but when there exists some $t \in S$ such that $t(r_1 s_2 - s_1 r_2)=0$.

Common misconceptions

• Thinking two fractions in $S^{-1}R$ are equal exactly when $r_1s_2=s_1r_2$.
• Assuming the naive cross-multiplication rule is sufficient in the general localization setting.

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Connected concepts

Explore next

• Why is the naive cross-multiplication condition insufficient in the general localization setting?
• What is the equivalence relation for fractions in ring localization?

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