Exercise 3.4

Question

Let $f\colon A 	o B$ be a homomorphism of rings and let $S$ be a multiplicatively closed subset of $A$. Let $T= f(S)$. Show that $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$ modules.

Amit Awasthi · Accepted Answer

\begin{proof}
Let $\phi$ be a natural $S^{-1}A$-module homomorphism from $S^{-1}B$ to $T^{-1}B$ given by $b/s \mapsto b/f(s)$. As $f$ is a homomorphism of rings, $\phi$ is a homomorphism. Let $\psi: T^{-1}B	o S^{-1}B$ such that $\psi(b/f(s)) = b/s$ for each $b \in B, f(s) = T = f(s)$. It is immediate that $\phi$ and $\psi$ are inverses of each other. Hence, $S^{-1}B$ and $T^{-1}B$ are isomorphic as $S^{-1}A$ modules. 
\end{proof}

Exercise 3.4

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Exercise 3.4

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