Exercise 3.3

Question

Let $A$ be a ring, let $S$ and $T$ be two multiplicatively closed subsets of
  $A$, and let $U$ be the image of $T$ in $S^{-1}A$.
  Show that the rings $(ST)^{-1}A$ and $U^{-1}(S^{-1}A)$ are isomorphic.

Amit Awasthi · Accepted Answer

\begin{proof}
  Let $g\colon A 	o U^{-1}(S^{-1}A) \eqqcolon B$ be the canonical ring
  homomorphism $a \mapsto a/1$.
  To show $(ST)^{-1}A \cong U^{-1}(S^{-1}A)$, it suffices to show the
  properties in Cor.~3.2 hold.
  \par Suppose $st \in ST$ where $s\in S,t\in T$.
  Then, $g(st) = st/1 = (s/1)(t/1)$ is a unit in $B$ since
  $t/1 \in U$ and $s/1$ is a unit in $S^{-1}A$, hence $3.2i)$ holds.
  \par Suppose $g(a) = 0$. Then, $a/1 \equiv 0$ and so $(t/1)(a/1) = at/1 = 0$ in
  $S^{-1}A$ for some $t/1 \in U$, hence $ast = 0$ for some $st \in ST$,
  hence $3.2ii)$ holds.
  \par Now suppose we have a unit $(a/s)/(t/1)$ in $B$. We claim $(a/s)/(t/1) =
  g(a)g(st)^{-1}$. For, $g(a)g(st)^{-1} = (a/1)(st/1)^{-1} = (a/1)(1/st) =
  (a/1)/(st/1)$, which is equivalent to $(a/s)/(t/1)$ since $(a/1)(t/1) =
  (a/s)(st/1)$ in $S^{-1}A$.
  Thus, $3.2iii)$ holds, and so $(ST)^{-1}A \cong U^{-1}(S^{-1}A)$ by Cor.~3.2.
\end{proof}

Exercise 3.3

Answers

Comments

Exercise 3.3

Answers

Comments

Add answer