Definition: Subring

Let \((R, + ,\cdot)\) be a ring. A subset \(S\subseteq R\) is called a subring, if \((S, + ,\cdot)\) itself is a ring; or, equivalently1, if and only if for all \(a,b\in S\) the following conditions are fulfilled:


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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013

Adapted from CC BY-SA 3.0 Sources:

  1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück


  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.