Thanks Owen, but \\((\mathbb{Z}/2, \cdot)\\), which corresponds to the discussed example, when \\(f \circ f = f\\), seems to be a monoid, because there is no inverse for \\(f\\), right?

So \\((\mathbb{Z}/2, +)\\), \\((\mathbb{Z}/2, \cdot)\\) is a ring, but is called a field, and this is the source of my confusion :)