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Section Day 15
This is an outline of the topics we covered in the fifteenth 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 Thursday 7/9
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.
Perform computations in quotient rings, including identifying distinct cosets
State and instantiate the definition of: prime ideal, maximal ideal
State the following mathematical results: Quotient Characterization of Prime and Maximal Ideals
Algebraist of the Day.
David Eisenbud , 1947-present
American mathematician working in commutative algebra and algebraic geometry
Director of MSRI at Berkeley from 1997-2007 and 2013-2022, Professor at UC Berkeley
Author of the book
Commutative Algebra with a View Toward Algebraic Geometry , one of the most widely used books in commutative algebra
Two papers on the mathematics of juggling
Definition 157 . Prime and Maximal Ideals.
Let
\(R\) be a commutative ring. A proper ideal
\(I\) of
\(R\) is
prime if
\(ab\in I\) implies
\(a\in I\) or
\(b \in I\text{.}\) A proper ideal
\(I\) of
\(R\) is
maximal if the only ideal properly containing
\(I\) is
\(R\text{:}\) if
\(J\) is an ideal and
\(I \subseteq J \subseteq R\) then either
\(I=J\) or
\(J=R\text{.}\)
Example 158 . Prime and Maximal Ideals in \(\Z\) .
Example 159 . Prime and Maximal Ideals in \(\Z[x]\) .
Theorem 160 . Quotient Characterization of Prime and Maximal Ideals.
Let \(R\) be a commutative ring with \(1\) and \(I\) be an ideal of \(R\text{.}\) Then
\(R/I\) is an integral domain if and only if
\(I\) is prime.
\(R/I\) is a field if and only if
\(I\) is maximal.
Proof.
