Math 441 - Abstract Algebra - Schedule

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Problem 24 (A Matrix Factor Ring)

Consider the set of two by two matrices $R=M_2(\mathbb{Z})$ with integer coefficients. Let $A$ the set of matrices whose coefficients are multiples of 3.

Show that $A$ is an ideal of $R$.
Find three different elements of $R$ that are in the coset $\begin{bmatrix}5&-2\\7&9\end{bmatrix}+A$.
How many distinct elements are in this factor ring? Justify your answer.
Challenge: This factor ring is isomorphic to another matrix ring we have already seen. What is this ring? You don't need to prove your answer.

Problem 27 ($R/A$ Is An Integral Domain Iff $A$ is prime)

Let $R$ be a commutative ring with unity, and let $A$ be a proper ideal of $R$. Prove that the following are equivalent.

$R/A$ is an integral domain.
If $a,b\in R$ and $ab\in A$ then $a\in A$ or $b\in A$. (We say that $A$ is a prime ideal).

Why do we call an ideal a prime ideal if the second condition above is satisfied? Recall in the integers that if $p$ is a prime, then if $ab$ is a multiple of $p$, then either $a$ or $b$ must be a multiple of $p$. So $ab\in \left<p\right>$ forces either $a\in \left<p\right>$ or $b\in \left<p\right>$. This condition forces the number to be prime. We extend the notation of prime numbers to prime ideals. Any time a product is divisible by a prime, then one of the two factors must be divisible by the prime. We'll extend this to say that any time a product is in a prime ideal, one of the factors must be in the prime ideal.

We now have a simple way to check if $R/A$ is an integral domain. We just have to make sure that if $ab\in A$, then either $a\in A$ or $b\in A$. To obtain a field, we need even more.

Problem 28 (Characterizing When Factor Rings Are Fields)

Suppose $R$ is a commutative ring with unity and $A$ is an ideal. Prove that the following are equivalent.

$R/A$ is a field.
For every $b\in R$ with $b\notin A$, the set $B=\{bc+a\mid c\in R, a\in A\}$ contains the unity $1$ from $R$.

Problem 29 (Obtaining A New Ideal By Adding One Element)

Let $A$ be an ideal of the commutative ring $R$ with unity. We'd like to increase the size of $A$ by adding in a single element $b$. Show the following:

Pick $b\in R$. The set $B=\{bc+a\mid c\in R,a\in A\}$ is an ideal of $R$ that contains both the element $b$ and subset $A$.
We know $1\in A$ if and only if $A=R$.

The set $B$ described above is the smallest ideal of $R$ that contains $b$ and every element in $A$, so we call it the ideal generated by $b$ and $A$.

Given a proper ideal $A_1$, we can use the idea above to create an ideal $A_2$ that properly contains $A_1$ (add one element not in $A_1$). If this ideal is not the entire ring, we could then continue this process, possibly indefinitely, to obtain a sequence of ideals that properly contain the previous. Such a sequence $A_1\subsetneq A_2\subsetneq A_3\subsetneq \cdots \subsetneq A_n\subsetneq \cdots$ is called an ascending chain of ideals. Once one of these ideals contains the number 1, the sequence terminates. Emmy Noether used this idea to create what we now call Noetherian domains, an integral domain in which any ascending chain of ideals must be finite in length (so the process must stop, no matter what ideal $A$ you start with). We'll come back to Noetherian domains later. See page 275 for more information about Emmy Noether.

Problem 30 (An Ascending Chain Of Ideals In The Integers)

Let $R=\mathbb{Z}$. Consider the ideal $A_1=\left<56\right>.$

Find an ascending chain of ideals consisting of three ideals $A_1\subsetneq A_2\subsetneq A_3$ with $A_3=R$.
What is the largest $n$ such that $A_1\subsetneq A_2\subsetneq A_3\subsetneq \cdots\subsetneq A_n=R$ is an ascending chain of ideals? How many such chains are there of this length?
If $n\in \mathbb{Z}$, describe a process we could use to find a longest possible ascending chain of ideals with $\left<n\right>\subsetneq A_2\subsetneq \cdots\subsetneq A_n=\mathbb{Z}$.

The two facts from the problem Obtaining A New Ideal By Adding One Element will help us characterize the properties of an ideal $A$ that will result in $R/A$ being a field. Remember, the following:

We need to make sure that if $b\notin A$, then there exists $c\in R$ such that $bc\in 1+A$, or equivalently $1\in bc+A$.
So given any element $b\notin A$, if we can show that the ideal $B=\{bc+a\mid c\in R, a\in A\}$ must contain 1, which means $B=R$, then we'd win.
This is basically the same as showing that given any ideal $B$ trapped between $A$ and $R$, so $A\subseteq B\subseteq R$, then if $B$ is not equal to $A$, it must equal $R$. We call an ideal such as $A$ a maximal ideal.

The next problem has you carefully show that $R/A$ is a field if and only if $A$ is a maximal ideal.

Problem 31 ($R/A$ Is A Field Iff $A$ Is Maximal)

Let $R$ be a commutative ring with unity, and let $A$ be a proper ideal of $R$. Prove that the following are equivalent.

$R/A$ is a field.
Whenever $B$ is an ideal of $R$ and $A\subseteq B\subseteq R$, then either $B=A$ or $B=R$. (We say that $A$ is a maximal ideal.)

The previous problems give us the following definitions of prime and maximal ideals. Basically, these are now characteristics of an ideal that we can check to determine if $R/A$ is an integral domain or a field, without having to ever consider elements of the factor ring.

Definition (Prime Ideal And Maximal Ideal)

A prime ideal $A$ of a commutative ring $R$ is a proper ideal of $R$ such that $a,b\in R$ and $ab\in A$ imply $a\in A$ or $b\in A$.
A maximal ideal $A$ of a commutative ring $R$ is a proper ideal of $R$ such that, whenever $B$ is an ideal of $R$ and $A\subseteq B\subseteq R$, then $B=A$ or $B=R$.

Let's look at another example of some interesting things that can happen in a ring.

Definition (Idempotent And Nilpotent)

Let $R$ be a ring.

We say that $a$ is an idempotent of $R$ if $a^2=a$.
We say that $a$ is nilpotent if $a^n=0$ for some integer $n$.

Problem 32 (Finding Idempotent And Nilpotent Elements In A Matrix Ring)

Complete the following:

In the ring $M_2(\mathbb{Z})$, the matrices $\begin{bmatrix}0&0\\0&0\end{bmatrix}$, $\begin{bmatrix}1&0\\0&1\end{bmatrix}$, $\begin{bmatrix}1&0\\0&0\end{bmatrix}$, $\begin{bmatrix}0&0\\0&1\end{bmatrix}$, are idempotents. Find two others.
Show that $\begin{bmatrix}0&t\\0&0\end{bmatrix}$ is nilpotent in $M_2(\mathbb{Z})$ and that $\begin{bmatrix}0&t&0\\0&0&t\\0&0&0\end{bmatrix}$ is nilpotent in $M_3(\mathbb{Z})$, where $t\in\mathbb{Z}$.
Let $X=\begin{bmatrix}0&t&0&0&0\\0&0&t&0&0\\0&0&0&t&0\\0&0&0&0&t\\0&0&0&0&0\end{bmatrix}$. Compute $X^k$ for each positive integer $k$. What patterns do you see? Is $X$ a nilpotent element of $M_5(\mathbb{Z})$?

The matrices above are central to finding the Jordan form of a matrix, and then using the Jordan form to compute the matrix exponential. Come see me out of class if you would like more information on this topic.

For more problems, see AllProblems

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