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'ground fields and rings'
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| Title of object: |
ground fields and rings |
| Canonical Name: |
GroundFieldsAndRings |
| Type: |
Definition |
| Created on: |
2006-05-08 19:26:29 |
| Modified on: |
2006-05-10 01:34:59 |
| Classification: |
msc:08A30 |
| Keywords: |
ground ring |
| Defines: |
ground field, base field, ground ring, base ring |
| Synonyms: |
ground fields and rings=base field
ground fields and rings=base ring |
Revision comment (for changes between this and next version):
| changed the "now" to "no" -- a typo! also added an "a" were needed. |
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Content:
The following is a list of common uses of the term \emph{ground} and/or \emph{base} field and/or ring in algebra. These terms are endowed with semantics based on their context so the following list may be incomplete or may not apply uniformly.
One commonality is generally found for the use of ground ring/field: the result is a unitial subring of the original. Outside of this requirement, the constraints are specific to context.
\begin{itemize}
\item Given a ring $R$ with a 1, let $\mathbb{Z}1$ be the subgroup of $R$ generated by $1$ under addition. This is consequently a subring of $R$ of the same characterisitic as $R$. Thus is it isomorphic to $\mathbb{Z}/c\mathbb{Z}$ where $c$ is the characterisitic of $R$. This is the smallest unital subring of $R$ and so rightfully may be called the ground or base ring of $R$.
When the characteristic of $R$ is prime, $\mathbb{Z}1\cong \mathbb{Z}/p\mathbb{Z}$ and so it may be called the ground field of $R$.
\item Given an algebra $A$ over a field $k$, then $k$ is the ground/base field of $A$.
\item Given a set of matrices $M_n(R)$, the ground ring is commonly the ring $R$, and if required as a subring of $M_n(R)$ then it is taken as the set of all scalar matrices.
\item Given a field extension $K/k$ over a field $k$, then $k$ is the ground field of $K$ in this context. For a general field where now specific subfield has been specified, the ground/base field then typically defaults to the prime subfield of $K$. (Recall the prime subfield is the unique smallest subfield of $K$.)
\item Given a field $K$ and a set of field automorphisms\, $f:K\rightarrow K$,\, the ground/base field in this context is the fix field of the automorphisms. That is, the largest subfield of $K$ which is pointwise fixed by each $f$. Since field automorphism must fix the prime subfield, this definition always produces a field containing the prime subfield.
\end{itemize}
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