Using the Chinese Remainder Theorem

#ring_theory

(a) 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 / I J ≅ R / I \times R / J .

(b) Prove that $(Z / 3 Z) [x] / (x^{3} - x^{2} - 1) ≅ (Z / 3 Z) [x] / (x^{3} + x + 1)$ . (Hint: Use part (a).)

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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}