Ring extension

Description

Like the Field extension is the Ring extension a pair of consisting of an Extension ring and a Subring. Note that the Ring extension is not the same as the Extension ring. The latter more heavily focuses on the bigger ring whereas the former weighs both rings equally. We use the word ring extension often when we are talking about Automorphismus, fixing the subring.

Definition

Let $S,R$ be rings where $S\subseteq R$. Then the tuple $R\mid S$ is called a ring extension.

Another use of the word ring extension is the following:

Definition

Let $\tilde{R}\mid R$ be a ring extension ($\tilde{R}$ is a context ring, e.g. the Algebraischer Abschluss) and let $A\subseteq \tilde{R}$ be a subset. Then there is is a unique smallest subring of $\tilde{R}$ containing both $R$ and $A$. See Generated subring for more info.