Generated subring

Description

The generated subring is the smallest extension of a ring by some number. It is used to easily generate interesting examples of rings or it is used heavily in Galois theory.

Definition

Let $\tilde{R}|R$ be a Ring extension and $A\subset \tilde{R}$ be a subset. Then there is a unique Subring $R[A]$ of $\tilde{R}$ fulfilling the following properties:

1. It contains $R$ and $A$: $R[A]\supset R\cup A$
2. It is the smallest possible subring: If $R^{\prime }$ is another subring of $\tilde{R}$ fulfilling $R^{\prime }\supset R\cup A$, then $R^{\prime }\supset R[A]$

The elements $A$ are seen to be taken out a “context” $\tilde{R}$.

Notation: If $A$ consists of one element, we often drop the parentheses.

Properties

Explicit description of elements

Let $\tilde{R}|R$ be a Ring extension and $c\in \tilde{R}$. The elements in $R[c]$ may be described as follows: $R[c]=\left\{{\sum }_{i=0}^n{a}_i{c}^i:n\in {\mathbb{N}}_0,a_0,...,a_n\in R\right\}$ i.e. they are polynomials in $c$.

Examples

Quadratic extensions

See Quadratische Zahlringe

Gaussian numbers

The Gausssian numbers are an important ring. They are given by $\mathbb{Z}[i]$

Eisenstein numbers

The Eisenstein numbers are given by $\mathbb{Z}[\frac{1}{2}(1+\sqrt{-3})]$. They are interesting as they form a triangular lattice.