Let \((R, + ,\cdot)\) be a ring. A subset \(S\subseteq R\) is called a subring, if \((S, + ,\cdot)\) itself is a ring; or, equivalently^{1}, if and only if for all \(a,b\in S\) the following conditions are fulfilled:
equivalenty^{1}
Please note that both definitions are indeed equivalent. For if \((S, + ,\cdot)\) is a ring, then (S1) and (S2) are trivially fulfilled. On the other side, if (S1) is fulfilled, then following the subgroup definition \((S, +)\) is a commutative subgroup of the additive commutative group \((R, + )\). Correspondingly, if (S2) is fulfilled, then \(S\) is closed under the multiplication, and therefore \((S,\cdot)\) is a monoid, since \((R,\cdot)\) is. The distributive law is then inherited by \(S\) from \(R\). Since all defining conditions of a ring are fulfilled for \(S\), \((S, + ,\cdot)\) is a ring. ↩↩