Do countable unital rings with uncountably many distinct right ideals have uncountably many maximal right ideals? Suppose we are given a countable unital ring $R$ with uncountably many distinct right ideals. Does it follow from this that $R$ has uncountably many maximal right ideals?

Let $V$ be a $\mathbb Q$-vector space of countable dimension and let $R=\mathbb Q\oplus V$ with commutative multiplication such that the injection $\mathbb Q\to R$ is a map of rings, multiplication between $\mathbb Q$ and $V$ is the obvious one, and $v\cdot w=0$ for all $v$, $w\in V$.
Every subspace of $V$ is an ideal of $R$, so there are uncountably many of these, yet $R$ is local.
