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I'm assuming the idea behind the problem is to prove that the additive identity and multiplicative identity are the same.This can only happen if either 1 or 0 or both are part of the Ring.

If R={1},then all the axioms that define a ring are satisfied.

If R={0},then again all the axioms that define a ring are also satisfied.....and if If R={0,1} it is the same thing so in order for 1_r=0_r the Ring must have one element which is either 0 or 1...is this correct?