How to check a set of ring is a subring? To check a subset of a given ring is a subring,
is it enough to check that the subset is closed under induced operations(multiplication and addition) or
do I also need to show that it contains 0 and additive inverses of each element?

If you're interested in a single additive property to check, it suffices for the subset to be closed under subtraction and multiplication. Subtraction gives you $0$ by $a-a$, the additive inverse by $0-a$ and thus addition by $a-(-b)$. In practice this isn't really easier than just checking addition and negation, however.
