Why is there no subring of the integers modulo 29 isomorphic to the integers modulo 8?

Question

I understand that for rings R and S, S contains a subring isomorphic to R if and only if there is an injective ring homomorphism from R to S. But I am unsure of how to proceed from this definition?

Also that for rings to be isomorphic you would instead need a bijection.

Any help would be appreciated, thanks!

Answer 1

Hint: The integers mod $29$ is a field, and the integers mod $8$ has nonzero zero divisors.

Answer 2

Hint: the additive group of the intergers modulo $29$ is cyclic of prime order and has no proper non-trivial subgroup.