Algebra by Michael Artin Ch3, Ch11
Artin has different definitions of rings particularly that his rings are commutative in both addition and multiplication. Based on his definitions, (*) I believe that $0$ is in subrings for the same reason $0$ is in subfields: (**) $$1-1=0$$
Am I mistaken?
(*)
Definition of a subring of the ring of complex numbers $\mathbb C$ (and then I guess this is extended to a subring of a ring $R$)
Definition of a ring
(**)
Earlier, subfields of the field $\mathbb C$, fields and subfields of fields were defined similarly.
Definition of a subfields of the field of complex numbers $\mathbb C$ (and then I guess this is extended to a subfield of a field $F$)
Definition of a field
Subrings contain $0$ because they are, in particular, groups (written additively). Recall that a ring is a group written additively with a mutliplicative structure. So, not all rings have 1 but all rings have 0 since that is the identity element in the underlying group.
So, why does a subring contain 0? Because it is a ring and so, in particular, an abelian group written additively. Hence, contains an identity which in this case would be 0.