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We are given the ideal $(\bar{3})$ in $\mathbb{Z}_9$, and we are to check whether it is maximal. --- ### Step 1: Understanding the Ideal We are working in the ring $\mathbb{Z}_9 = \mathbb{Z}/9\mathbb{Z}$, which is the set $\{ \bar{0}, \bar{1}, \bar{2}, \ldots, \bar{8} \}$ with addition and multiplication modulo 9. The ideal $(\bar{3})$ is the principal ideal generated by $\bar{3}$ in $\mathbb{Z}_9$. That means: $$ (\bar{3}) = \{ \bar{3} \cdot \bar{x} \mid \bar{x} \in \mathbb{Z}_9 \} $$ So, ____ __________ _____ _________ ____ ___.
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