Find all prime ideals and all maximal ideals of Z2 * Z4

Ideals of Z₂ x Z₄ must have order dividing 8.

Since (Z₂ x Z₄) / (Z₂ x Z₄) ≅ {0}, there is no maximal/prime ideal of order 8.

Order 4:
{0} x Z₄: Note that (Z₂ x Z₄) / ({0} x Z₄) ≅ Z₂ is a field/integral domain.
Z₂ x 2Z₄: Note that (Z₂ x Z₄) / (Z₂ x 2Z₄) ≅ Z₂ is a field/integral domain.
Hence, these two ideal are both prime and maximal.

Order 2:
{0} x 2Z₄:Note that (Z₂ x Z₄) / ({0} x 2Z₄) ≅ Z₂ x Z₂ is not even an integral domain
Z₂ x {0}: Note that (Z₂ x Z₄) / (Z₂ x {0}) ≅ Z₄ is not even an integral domain.
So, these are not prime or maximal ideals.

I hope this helps!