In article . com "Schoenfeld" writes:
....
That looks reasonable, but there is a subtely here which I can't deduce
from the axioms.

The axiom stated:
There exists y such that for all x, x + y = x

Does this mean that,
STATEMENT 1:
For all y in set of additive identities, for all x in Z, x + y = x.

You make this assumption when you state that:
x + Y1 = x
x + Y2 = x

We have the ring axiom that there is a 0 such that for all x: x + 0 = x.
We have the ring axiom that for each x there is a (-x) such that x + (-x) = 0.
We have also commutativity and associativity of the addition.

Now suppose there is an y such that x + y = x for some x. Add (-x) to both
sides and find that y = 0:
y = // by y + 0 = y
y + 0 = // commutativity
0 + y = // x + (-x) = 0
x + (-x) + y = // commutativity
(-x) + x + y = // by x + y = x
(-x) + x = // x + (-x) = 0
0
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