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well, we must impose restrictions on the ring $R$. If $R$ is finite maximal subrings mean subring of maximal order. If $R$ is infinite suppose that $R$ is a principal ideal domain.
I think that a non-free subalgebra is never maximal, because is contained in a free algebra. For example in the example of Florian Eisele if we take $b\in R$ (and not in the ideal I) the subring is free.
