2 moved the question to the title

# LargeAretherecountableindex subrings of the realnumbersreals?

1. Does ${\mathbf {\mathbb R}$ have proper, countable index proper subrings? By countable I mean finite or countably infinite. By subring I mean any additive subgroup which is closed under multiplication (I don't care if it contains $1$.) By index, I mean index as an additive subgroup.

2. Given some real number $x$, when is it possible to find a countable index subring of ${\mathbf {\mathbb R}$ which does not contain $x$?

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# Large subrings of the real numbers

1. Does ${\mathbf R}$ have countable index proper subrings? By countable I mean finite or countably infinite. By subring I mean any additive subgroup which is closed under multiplication (I don't care if it contains $1$.) By index, I mean index as an additive subgroup.

2. Given some real number $x$, when is it possible to find a countable index subring of ${\mathbf R}$ which does not contain $x$?