# 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:

• (S1) $$a-b\in S$$.
• (S2) $$a \cdot b\in S$$.

equivalenty1

• $$(S, + )$$ is a subgroup of $$(R,+)$$ and
• $$(S,\cdot)$$ is a submonoid of $$(R,\cdot)$$.

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### References

#### Bibliography

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

#### Footnotes

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.