TITLE: How to check a set of ring is a subring?
QUESTION [7 upvotes]: 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?

REPLY [8 votes]: Let's denote this set by $S$. It's a subring if:

$S\ne \emptyset$ and in practice we prove that $0\in S$;
$\forall a,b\in S$, $a-b\in S$ that's $S$ is a subgoup;
$\forall a,b\in S$, $ab\in S$.