 Solution

$\mathbb{Z}/4\mathbb{Z}$ is a ring but not a field. Quotient operations inherit associativity and distributivity from the integers, and residues have additive inverses. However the nonzero residue $\overline{2}$ satisfies $\overline{2}\overline{2}=\overline{0}$, so it cannot be invertible. A field has no nonzero zero divisors.