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every element in S will have unique identity and inverse,

for identity, e=0

x*0=x  { x+0+0*x = x  }

for inverse,

let y be inverse of x

x*y=0

x+y+xy=0

y=-x/(1+x)   { 1+x !=0)

x!= -1
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Yes, S is the subgroup closed under * and it does show the abelian property. Remember its mentioned that s is improper subset of R, thus it should satisfy the property for the most.