We prove an exact formula for the asymptotic dimension (asdim) of a free product. Our main theorem states that if A and B are finitely generated groups with asdim A = n and asdim B ≤ n, then asdim (A*B) = max n,1.
We investigate conditions on an infinite simple group in order to construct a zero-symmetric nearring with identity on it. Using the Higman-Neumann-Neumann extensions and Clay’s characterization, we obtain zero-symmetric nearrings with identity with the additive groups infinite simple groups. We also show that no zero-symmetric nearring with identity can have the symmetric group as its additive group.