A subset is a subring exactly when it is an additive subgroup and multiplicative submonoid.Suggest a slogan
Let \(R\) be a ring. A subset \(S\subseteq R\) is a subring of \(R\) if and only if
\((S, +)\) is a subgroup of \((R,+)\): \(0\in S\), and \(a-b\in S\) for all \(a,b\in S\); and
\((S, \cdot)\) is a submonoid of \((R,\cdot)\): \(1\in S\), and \(ab\in S\) for all \(a,b\in S\).