Let $R$ be a ring with fraction ring $R_S$ and ideal $I$. I saw in arguments that when $a/s$ is in $I_S$ they dont say $a$ is in $I$. Instead they say $a/s=b/t$ with $b \in I$. Why?
Many thanks.
2026-03-30 04:44:08.1774845848
Fraction rings ideals members
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From Steps in Commutative Algebra by Sharp:Consider the ring $\mathbb Z_3$ of fractions of $\mathbb Z$ with respect to the multiplicatively closed subset $\{3^i : i \in \mathbb N_0 \}$ of $\mathbb Z$. Set $I = 6\mathbb Z.$ Then $6/3^2\in S^{-1} I$, so $2/3\in S^{-1} I.$ But $2\notin I.$