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Section Day 13
This is an outline of the topics we covered in the thirteenth 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 6/30
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 instantiate the definition of: divides, factor, subring, zero-divisor, integral domain
State the following mathematical results: Ring Multiplication Rules, Uniqueness of Unity and Inverses
State and apply the Subring Test to determine if subsets of a ring are subrings
State the following mathematical result: Integral Domains have Cancellation
Algebraist of the Day.
William Hamilton , 1805-1865
Irish mathematician, physicist, and polyglot.
Mastered 14 languages by age 14
First modern treatment of complex numbers, extended to the quaternions in 1843 (three imaginary units instead of one; first noncommutative ring discovered)
Also known for the Hamiltonian function in physics and the Cayley-Hamilton theorem in linear algebra. Coined the words
vector, scalar, tensor
Definition 136 . Subring.
A subset
\(S\) of a ring
\(R\) is a
subring of
\(R\) if
\(S\) is a ring with the operations of
\(R\text{.}\)
Example 137 . Subring Example.
Theorem 138 . Subring Test.
A nonempty subset \(S\) of a ring \(R\) is a subring of \(R\) if and only if \(S\) is closed under subtraction and multiplication:
\begin{equation*}
a-b,ab \in S
\end{equation*}
whenever \(a,b \in S\text{.}\)
Example 139 . Using the Subring Test.
Let
\(a\in \Z\) and consider
\(S=\{f\in \Z[x]\mid f(a)=0\}\text{.}\)
Definition 140 . Zero-Divisor.
Let
\(R\) be a commutative ring. An element
\(a\in R\setminus\{0\}\) is a
zero-divisor if there is a nonzero
\(b \in R\) such that
\(ab=0\text{.}\)
Definition 141 . Integral Domain.
A commutative ring with identity is an
integral domain if it has no zero-divisors.
Example 142 . Zero-Divisors in \(\Z_n\) .
Example 143 . Zero-Divisors in Infinite Rings.
Proposition 144 . Integral Domains Have Cancellation.
Let
\(R\) be an integral domain. If
\(a,b,c\in R\) and
\(a\neq 0\text{,}\) then
\(ab=ac\) implies
\(b=c\text{.}\)
Proof.
