3
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Let $R$ be the ring obtained by taking the quotient of $\mathbb{Z_6}[X]$ by principal ideal $(2X+4)$. Then

1) $R$ has infinite elements

2) $R$ is field

3) $5$ is unit in $R$

4) $4$ is unit in $R$.

My Attempt: \begin{equation*} \mathbb{Z_6}[X]/(2X+4) = \mathbb{Z_6}[X]/(2(X+2)) = \mathbb{Z_3}[X], \end{equation*} as $X+2$ has root in $\mathbb{Z_6}$. According to me 1) and 3) are correct but I'm not sure about my answer. Help me. I am not more familiar with polynomial and quotient ring. I searched this problem here but couldn't get this here. If this problem is already asked here then how can I get this?

Thank you in advance.

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  • $\begingroup$ Your isomorphism is wrong, but 1) and 3) are correct. $\endgroup$ – user26857 May 24 '15 at 9:50
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$\mathbb{Z_6}[X]/(2X+4)\simeq\mathbb{Z_2}[X]/(2X+4)\times\mathbb{Z_3}[X]/(2X+4)\simeq\mathbb{Z_2}[X]\times\mathbb{Z_3}$

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