Let S be a non-empty subset of a ring R. Then S is a subring of R if and only if S is closed under $-$ and $\times$.
Proof: First, prove that S is a subgroup of R. Pick an arbitrary element $x$ from S. Since S is closed under subtraction, $x - x = 0 \in S$ and $0 - x = -x \in S$. Therefore, for all $x, y \in S, -x, -y \in S$ and hence $x-(-y) = x+y$ also belongs to $S$. Hence S is a subgroup.
Since R is a ring, distributive laws hold in R and hence also in S. Abelian property is also inherited from R.
S is closed under multiplication.
Hence, S is a subring.
Is my proof OK?