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Section Day 16
This is an outline of the topics we covered in the sixteenth 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 7/14
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 and apply the following mathematical results: Properties of Ring Homomorphisms, Kernels are Ideals, Ideals are Kernels
State and apply isomorphism theorems for rings, especially the First Isomorphism Theorem
State and apply the following mathematical results: Natural Homomorphism from
\(\Z\) to a Ring with Identity and Corollaries (Ring with Identity contains
\(\Z_n\) or
\(\Z\text{,}\) Fields Contain
\(\Z_p\) or
\(\Q\) )
Algebraist of the Day.
Joseph Gallian , 1942-present
Professor at University of Minnesota Duluth
Work in group theory and combinatorics, including the current major survey of graph labeling results
President of MAA 2007-2009, co-director of Project NExT 1998-2012, ran REUs at U M-D for 46 years
Determined the mathematical method used by Minnesota to assign drivers’ license numbers
Taught a course called "The Lives and Music of Beatles" for 33 years
Proposition 163 . Ring Homomorphism Properties.
Let \(\phi:R \to S\) be a ring homomorphism, \(A\) be a subring of \(R\text{,}\) and \(I\) be an ideal of \(S\text{.}\) Then
For \(r\in R, n \in \Z_{\gt 0}\text{:}\)
\begin{equation*}
\phi(\sum_{i=1}^n r) = \sum_{i=1}^n \phi(r) \quad \text{and} \quad \phi(r^n)=\phi(r)^n
\end{equation*}
\(\phi(A)\) is a subring of
\(S\)
If
\(A\) is an ideal of
\(R\) and
\(\phi\) is onto, then
\(\phi(A)\) is an ideal of
\(S\)
\(\phi^{-1}(I)=\{r\in R \mid \phi(r)\in I\}\) is an ideal of
\(R\)
\(R\) is commutative implies
\(\phi(R)\) is commutative
If
\(R\) has an identity
\(1_R\) and
\(\phi\) is onto, then
\(S\) has an identity and
\(1_S=\phi(1_R)\)
\(\phi\) is an isomorphism if and only if
\(\phi\) is onto and
\(\ker(\phi)=\{0_R\}\)
\(\phi\) is an isomorphism from
\(R\) to
\(S\) if and only if
\(\phi^{-1}\) is an isomorphism from
\(S\) to
\(R\)
Theorem 164 . Kernels are Ideals.
Let
\(\phi:R \to S\) be a ring homomorphism. Then
\(\ker(\phi)\) is an ideal of
\(R\text{.}\)
Theorem 165 . Ideals are Kernels.
Let
\(I\) be an ideal of a ring
\(R\text{.}\) Then
\(I\) is the kernel of the natural map
\(R\to R/I\) taking
\(r\mapsto r+ I\text{.}\)
Theorem 166 . First Isomorphism Theorem for Rings.
Let \(\phi:R \to S\) be a ring homomorphism. Then the map \(\Psi: R/\ker(\phi) \to \phi(R)\) given by
\begin{equation*}
\Psi(r+\ker(\phi))=\phi(r)
\end{equation*}
is a ring isomorphism. In short, \(R/\ker(\phi)\isom \im(\phi)\text{.}\)
Note 167 .
The Diamond, Fraction, and Correspondence Theorems also all carry over to rings, with normal subgroups replaced by ideals.
Theorem 168 . Natural Homomorphism from \(\Z\) to a Ring with 1.
Let \(R\) be a ring with identity \(1\text{.}\) The map \(\phi:\Z\to R\) given by
\begin{equation*}
\phi(n)=\sum_{i=1}^n 1:=n\cdot 1
\end{equation*}
is a ring homomorphism.
Proof.
Corollary 169 . A Ring with 1 Contains \(\Z\) or \(\Z_n\) .
Let
\(R\) be a ring with
\(1\text{.}\) If
\(\char(R)=n\gt 0\text{,}\) then
\(R\) has a subring isomorphic to
\(\Z_n\text{.}\) If
\(\char(R)=0\text{,}\) then
\(R\) has a subring isomorphic to
\(\Z\text{.}\)
Proof.
Corollary 170 . Every Field Contains \(\Z_p\) or \(\Q\) .
If
\(F\) is a field, then if
\(\char(F)=p\text{,}\) \(F\) contains a subfield isomorphic to
\(\Z_p\text{.}\) If
\(\char(F)=0\text{,}\) then
\(F\) contains a subfield isomorphic to
\(\Q\text{.}\) This subfield is called the
prime subfield of
\(F\text{.}\)
