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Section Daily Prep 15
You will practice with the definition of ideal and computation in quotient rings. We will also introduce the concepts of prime and maximal ideals.
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: ideal, quotient ring, principal ideal, ideal generated by elements
\(a_1,\dots,a_n\)
Perform computations in quotient rings, including identifying distinct cosets
Subsection Resources for Learning
Use these resources to prepare for class and answer the questions below.
Gallian, Chapter 14, pp. 267-270
Review the Day 14 notes as needed.
Figure 153. Reference Video for Quotients
Figure 154. Reference Video for Ideals
Subsection Important Terms
Theorem 155 . Ideal Test.
A nonempty subset \(I\) of a ring \(R\) is an ideal if we have
\(I\) is closed under subtraction:
\(a-b\in I\) for all
\(a,b \in I\)
\(I\) absorbs ring multiplication:
\(ar,ra\in I\) for all
\(a\in I\) and
\(r \in R\)
Definition 156 . Principal Ideal.
In a commutative ring \(R\text{,}\) the principal ideal generated by \(a\) is
\begin{equation*}
\ideal{a}=\{ra\mid r \in R\}\text{.}
\end{equation*}
