MHB Prove Ring with Identity on Set S with One Element x

[Stephen88]

On a set S with exactly one element x,
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?

 

[Evgeny.Makarov]

We have x + (-x) = 0 where both -x and 0 are defined to be x, so there is no problem with an additive inverse.

 

[Stephen88]

uh sorry...yes that is true then for multiplication commutativity (x*x)*=x*(x*x) and also x(x+x)=x +x and (x+x)x=x+x again.Will this suffice or is there something else.?..because it seemed quite short.

 

[Evgeny.Makarov]

for multiplication commutativity (x*x)*=x*(x*x)

The fact (x * x) * x = x * (x * x) is called associativity.

x(x+x)=x +x and (x+x)x=x+x again.

Distributivity says x(x+x) = x * x + x * x and (x+x) * x = x * x + x * x.

Will this suffice or is there something else.?..because it seemed quite short.

Why don't you check the list of ring axioms, for example, in Wikipedia?