Using the Chinese Remainder Theorem
-
Suppose
and are ideals in a commutative ring such that . Prove that the map given by induces the isomorphism -
Prove that
. (Hint: Use part (1).)
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\begin{enumerate}
\item Suppose $I$ and $J$ are ideals in a commutative ring $R$ such that $R=I+J$. Prove that the map $f:R\to R/I\times R/J$ given by $f(x)=(x+I,x+J)$ induces the isomorphism
\[
R/IJ\cong R/I\times R/J.
\]
\item Prove that $\left({\bf Z}/3{\bf Z}\right)[x]/(x^3-x^2-1)\cong \left({\bf Z}/3{\bf Z}\right)[x]/(x^3+x+1)$. ({\itshape Hint:} Use part (a).)
\end{enumerate}