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Section Day 14
This is an outline of the topics we covered in the fourteenth day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.
Handout Tuesday 7/7
Objectives: Advanced Learning Outcomes
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
State the following mathematical results: Characteristic of a Ring with Identity, Characteristic of an Integral Domain
State and instantiate the definition of: nilpotent, idempotent, ideal, quotient ring
Theorem 149 . Characteristic of a Ring with Identity.
Let
\(R\) be a ring with
\(1\text{.}\) If
\(1\) has infinite order under addition, then
\(\char(R)=0\text{.}\) If
\(1\) has order
\(n\text{,}\) then
\(\char(R)=n\text{.}\)
Proof.
Theorem 150 . Characteristic of an Integral Domain.
The characteristic of an integral domain is either
\(0\) or prime.
Proof.
Quotient Rings.
If \(A\) is a subring of a ring \(R\text{,}\) then we can form additive cosets of \(A\) in \(R\) as with groups,
\begin{equation*}
R/A = \{r+A\mid r \in R\}\text{.}
\end{equation*}
We have
\begin{equation*}
(r+A)+(s+A)=(r+s)+A
\end{equation*}
a well-defined operation since \((R,+)\) is Abelian. What about
\begin{equation*}
(r+A)(s+A) = rs + A\text{?}
\end{equation*}
Question: Is this a ring?
Definition 151 . Ideal.
An
ideal is a subring
\(I\) of a ring
\(R\) such that for any
\(a\in I\) and
\(r\in R\text{,}\) we have
\(ar,ra\in I\text{.}\)
Example 152 . Ideals and Non-Ideals.
