 With no generators the free monoid, free group, and free ring are all the one-element algebra consisting of just the additive identity 0. A ring with identity means having a multiplicative identity, that is, a word \(\varepsilon\). But this makes \(\varepsilon\) a generator for the additive group of the ring, and the free abelian group on one generator is the integers. So the free ring with identity on no generators is the integers under addition and now multiplication.