Subring

Description

A subring is a subset of a ring Ringes, which is also a ring and has the same 1-Element. This is the counterpart of a Extension ring.

Definition

Let $R$ be a Commutative Ring. A subset $S\subset R$ is called a Subring of $R$, if $1_R\in S$ holds and for $a,b\in S$ the elements $a-b$ and $ab$ lie in $S$.

Durch das $a-b$ spart man sich das Prüfen der Inversen und Abgeschlossenheit.

Properties

Intersection of subrings

Let $(R,+,\cdot )$ be a Ring, and let $(S_i{)}_{i\in I}$ be a family of Subrings. Then $S={\bigcap }_{i\in I}{S}_i$ is a subring of $R$

From this follows trivially that the Prime Field is a Subfield.