I have read earlier that in a ring $(R,+,.)$ the following needs to hold:
- $(R,+)$ is an abelian group
- multiplication is associative and closed
- left and right distribution laws hold.
However, I recently came across the fact that every ring has to have a multiplicative identity. Can anyone please clarify this? Is it needed for the ring to have a multiplicative identity?
(In fact it was mentioned that it is one of the reasons why $ker(f)$ is not a subring where $f$ is a ring homomorphism as the additive identity and the multiplicative identity are not usually in the same subset.)
Further in 2 different places I have noticed that there is a difference on whether the mapping $f(1) \to 1$ is a necessary condition for $f$ to be a ring homomorphism. I think this is also related to my doubt as to whether the multiplicative identity is in fact a necessary condition for defining a ring.