Ring Homomorphism and Group Homomorphism

Question

I'm new to ring, and I'm currently learning ring homomorphism. From my understanding, it is very similar to group homomorphism as it preserver operation. I wondered if some of the properties (in particular property 2) of group homomorphism still work for ring homomorphism.

For instance, does property 2 still applies to rings? Can I assume that if a map φ: R → S exists and R is a division ring or a field (has a multiplicative inverse), I can write something like this φ(g^−1) = [φ(g)]^−1?

Or I need to define the map to be onto and R a division ring or a field, then use property 7, which implies that eR map to eS, then I can use property 2.

Properties for group homomorphism that I learned:

• 1. If e is the identity of G1, then φ(e) is the identity of G2
• 2. For any element g ∈ G1, φ(g^−1) = [φ(g)]^−1
• 3. If H1 is a subgroup of G1, then φ(H1) is a subgroup of G2
• 4. If H2 is a subgroup of G2, then φ^−1(H2) = { g ∈ G1:φ(g) ∈ H2} is a subgroup of G1. Furthermore, if H2 is normal in G2, then φ−1(H2) is normal in G1

Properties for ring homomorphism that I learned so far:

• 5. If R is a commutative ring, then φ(R) is a commutative ring.
• 6. φ(0) = 0
• 7. Let 1R and 1S be the identities for R and S, respectively. If φ is onto, then φ(1R) = 1S
• 8. If R is a field and φ(R)̸={0}, then φ(R) is a field

Thank you,