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Ring with identity

StefanM

New member
Jan 30, 2012
28
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

Well-known member
MHB Math Scholar
Jan 30, 2012
2,488
We have x + (-x) = 0 where both -x and 0 are defined to be x, so there is no problem with an additive inverse.
 

StefanM

New member
Jan 30, 2012
28
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

Well-known member
MHB Math Scholar
Jan 30, 2012
2,488