环$C_2\times C_2$

hbghlyj

写出$R_1\to R_2$环同构
$$R_{1}=\frac{\mathbb{Z}_{2}[x]}{\left\langle x^{2}\right\rangle}, \quad R_{2}=\frac{\mathbb{Z}_{2}[x]}{\left\langle x^{2}+1\right\rangle}$$
Example 59只有提示没有写出来？
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hbghlyj

对于任何环$R$, $R[x]\to R[x];f(x)\mapsto f(x+1)$ 是环同构.
考虑$R=\Bbb Z_2[x],\bar\phi:\Bbb Z_2[x]\to\Bbb Z_2[x];f(x)\mapsto f(x+1)$
$\bar\phi(x^2)=(x+1)^2=x^2+1\bmod2$
这导致了环同构$\phi:\Bbb Z_2[x]/\langle x^2\rangle\to\Bbb Z_2[x]/\langle x^2+1\rangle;[f(x)]\mapsto[f(x+1)]$.

Czhang271828

小技巧:
\[
\dfrac{\mathbb{Z}_{2}[x]}{\left\langle x^{2}\right\rangle}\simeq\dfrac{\mathbb{Z}_{2}[x-1]}{\left\langle x^{2}\right\rangle}\simeq \dfrac{\mathbb{Z}_{2}[x]}{\left\langle (x+1)^{2}\right\rangle}=\dfrac{\mathbb{Z}_{2}[x]}{\left\langle x^{2}+1\right\rangle}.
\]