> A simple example where 1 + 1 + 1 = 1 is ℤ₂, the group of integers modulo 2 under addition.

That’s a good example of an algebra where 1 + 1 + 1 = 1, but the article is specifically about systems where in addition to that condition, this second condition is also true: 1 + 1 != 0 (not equal!). ℤ₂ is not an example of that.