Math 441 - Abstract Algebra - Schedule

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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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