Math 441 - Abstract Algebra - Schedule

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Let's now compare the field properties to operations with matrices and vectors. In the problems below, you'll be asked to list the properties of a field that are not satisfied. There are 9 properties to consider in each case, as there are 4 from each Abelian group and one more from the distributive laws. You should be prepared to prove each property that is satisfied (or at least state why you know it is satisfied).

Problem 5 (Algebraic Properties Of Square Matrices)

Let $\text{M}_2(\mathbb{Q})$ bet the set of 2 by 2 matrices with entries in $\mathbb{Q}$, together with the usual properties of matrix addition and matrix multiplication. Recall that $\text{GL}(2,\mathbb{Q})$ is the set of invertible 2 by 2 matrices with entries in $\mathbb{Q}$.

Which of the properties of being a field does $\text{M}_2(\mathbb{Q})$ not satisfy? Give an example of each property that is not satisfied.
Which of the properties of being a field does $\text{GL}(2,\mathbb{Q})$ not satisfy? Give an example of each property that is not satisfied.

Problem 6 (Algebraic Properties Of 3 D Vectors And The Cross Product)

Consider the set $\mathbb{R}^3$, together with the two binary operations of vector addition and the cross product. Which of the properties of being a field does $\mathbb{R}^3$ not satisfy? Give an example of each property that is not satisfied.

For fun, please see the wikipedia page on magmas. People have given lots of names to algebraic structures that satisfy various properties.

Definition (Ring)

A ring $R$ is an Abelian group $(R,+)$ together with an additional associative binary operation (multiplication) that satisfies the left and right distributive laws, namely $a(b+c)=ab+ac$ and $(b+c)a=ba+ca$.

Definition (Commutative Ring)

A commutative ring is a ring in which $ab=ba$ (multiplication commutes).

Definition (Unity And Unit)

A unity in a ring is a nonzero element that is an identity under multiplication. A nonzero element of a ring does not need to have a multiplicative inverse. When it does, we say that element is a unit of the ring.

Problem 7 (Recognizing Rings)

Which of the following are rings? Which rings have a unity? Which are commutative?

For each $n$, the set $\mathbb{Z}_n$ together with modular addition and multiplication.
The set $2\mathbb{Z}$ together with regular addition and multiplication.
The set $\mathbb{Q}[x]$ of all polynomials in the variable $x$ with coefficients in $\mathbb{Q}$ together with polynomial addition and multiplication.
Let $M_2(\mathbb{Z})$ bet the set of 2 by 2 matrices with entries in $\mathbb{Z}$, together with the usual properties of matrix addition and matrix multiplication.
Consider the set $\mathbb{R}^3$, together with the two binary operations of vector addition and the cross product.
The set of all real valued functions $f:\mathbb{R}\to\mathbb{R}$, together with function addition $(f+g)(x)=f(x)+g(x)$ and function multiplication $(fg)(x)=f(x)g(x)$.

Problem 8 (Rules Of Multiplication)

Let $R$ be a ring. Prove each of the following:

$a0=0a=0$
$a(-b)=(-a)b=-ab$
$(-a)(-b)=ab$
$a(b-c)=ab-ac$.
If $R$ has a unity, then $(-1)a = -a$.
If $R$ has a unity, then $(-1)(-1)=1$.

Definition (Subring)

A subset $S$ of a ring $R$ is a subring of $R$ if $S$ is itself a ring with the operations of $R$.

Problem 9 (Subring Test)

Let $R$ be ring. Prove that a nonempty subset $S$ of a ring $R$ is a subring of $R$ if $S$ is closed under subtraction and multiplication - that is, if $a-b$ and $ab$ are in $S$ whenever $a$ and $b$ are in $S$.

Problem 10 (Examples Of Rings Different Than The Integers)

For each item below, give an example of a ring $R$ and elements in the ring that satisfy the requested property.

$ab=0$ but neither $a$ nor $b$ equals $0$.
$ab=ac$ and $a\neq 0$ but $b\neq c$.
$ab=0$ but $ba\neq 0$.
$a^2=a$ but $a$ is neither 0 or 1.

Definition (Zero Divisor)

A zero-divisor is a nonzero element $a$ of a commutative ring $R$ such that there is a nonzero element $b\in R$ with $ab=0$.

Definition (Integral Domain)

An integral domain is a commutative ring with unity and no zero-divisors.

Problem 11 (Integral Domains Have The Cancellation Law)

Let $R$ be a commutative ring with unity. Prove that the following two statements are equivalent:

The ring $R$ is an integral domain.
For every $a,b,c\in R$ with $a\neq 0$, the equality $ab=ac$ implies $b=c$.

Definition (Gaussian Integers)

The set $\mathbb{Z}[i]=\{a+bi\mid a,b\in\mathbb{Z}\}$ is called the Gaussian integers.

Problem 12 (The Gaussian Integers Is An Integral Domain)

Use the subring test to show that the Gaussian Integers is a subring of the complex numbers $\mathbb{C}$. Then show that the Gaussian integers is an integral domain.

For more problems, see AllProblems

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