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define x + x = x, x*x = x. Prove that S is a ring.

The way I think about this problem is be showing that it verifies certain axioms....like associativity,commutativity,identity,inverse for addition and commutativity for multiplication and a (b + c) = ab + ac .. (a + b) c = ac + bc.

For Addition the first two i think it is obvious since

1.x+x=x+x..

2.(x+x)+x=x+(x+x)

For Identity since x+x=x then 0_S=x.

For the inverse I don't see how since the set has only one element x which equal 0_S....I guess I don't have to check the last two axioms because S is not a ring.

Am I doing this right?