Why is the naive cross-multiplication condition insufficient in the general localization setting?

Conditions

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

Reasoning, step by step

1. Recognize that in an integral domain, $r_1s_2 = s_1r_2$ is sufficient for fraction equality.
2. Note that in a general ring, $r_1s_2 - s_1r_2$ might not be zero, but could be annihilated by some $t \in S$.
3. Understand that the equivalence relation must allow for this annihilation to be well-defined and transitive.
4. Conclude that the condition $\exists t \in S : t(r_1s_2 - s_1r_2) = 0$ is necessary.

Example

The speaker states that equality holds "not only" when the cross product is zero, but also when multiplication by some $t \in S$ makes it zero. The board upgrades $(r_1 s_2 - s_1 r_2)=0$ to $\exists t \in S : t(r_1 s_2 - s_1 r_2)=0$.

Common misconceptions

• Mistaking the integral-domain rule for the general localization rule.
• Assuming that $r_1s_2 = s_1r_2$ is always the correct criterion for equality in $S^{-1}R$.

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Connected concepts

Explore next

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

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