Is there example of uncountable ring $R$ (with 1) containing a countable maximal left ideal $I$ .if yes , is answer unique ?
2026-03-30 05:15:29.1774847729
Is there example of uncountable ring $R$ (with 1) containing a countable maximal left ideal $I$?
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There are several such rings, for instance $R=\Bbb R\times \Bbb Z$, with the ideal $I=\{0\}\times\Bbb Z$. However, an uncountable ring $R$ with $1$ (but presumably even without $1$) can never have two countable maximal left ideals, because then by maximality $R\subseteq \mathfrak m_1+\mathfrak m_2$, and therefore the map $+:\mathfrak m_1\oplus \mathfrak m_2\to R$ would be surjective, which is impossible by cardinality.