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Section Daily Prep 14
You will learn more consequences of the definition of integral domain, the definition of the characteristic of a ring, and practice doing computations with both ideas.
Objectives: Basic Learning Objectives
Before our class meeting, you should use the resources below to be able to learn the following. You should be reasonably fluent with these; we’ll answer some questions on them in class but not reteach them in detail.
State and instantiate the definition of: integral domain, field, characteristic
State the following mathematical result: Finite Integral Domains are Fields
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 13, pp. 257-258
Figure 146. Reference Video for Integral Domains and Characteristic
Subsection Important Terms and Results
Theorem 147 . Finite Integral Domains are Fields.
Let
\(R\) be a finite integral domain. Then
\(R\) is a field.
Definition 148 . Characteristic of a Ring.
The characteristic of a ring \(R\text{,}\) denoted \(\char(R)\text{,}\) is the least positive integer \(n\) such that
\begin{equation*}
n \cdot x = 0
\end{equation*}
for all \(x \in R\text{,}\) if such an \(n\) exists. If no such \(n\) exists, we say \(R\) has characteristic \(0\text{.}\)
