MTH2001 Algebraic Structures Practical 1

• Practical Sheet.
• Practical Solutions.

Question 6.1

Given the set of $\mathbb{Z}/4\mathbb{Z}=\left\{0,1,2,3\right\}$…

• $2\cdot 2=0$. Hence the zero divisor is $\boxed{2}$.

Given the set of $\mathbb{Z}/12\mathbb{Z}=\left\{0,1,2,3,4,5,6,7,8,9,10,11\right\}$…

• $3\cdot 4=4\cdot 3=0$.
• $2\cdot 6=6\cdot 2=0$. Hence the zero divisors are $\boxed{2,3,4,6}$.

INCOMPLETE ANSWER 🕵

Also has 8,9,10 in the latter half, and all of those numbers should have lines above them to show that they’re congruence classes.

Question 6.2

Considering the rings $\mathbb{Z}$ and $\mathbb{Z}\times \mathbb{Z}$…

• $\boxed{\mathbb{Z}\text{ has unity }1}$ and $\boxed{\mathbb{Z}\times \mathbb{Z}\text{ has unity }(1,1)}$.
• $\mathbb{Z}$ has $\boxed{\text{no zero divisors}}$ and $\mathbb{Z}\times \mathbb{Z}$ has zero divisors of the form $\boxed{(a,0),(0,b):a,b\ne 0}$.
• Definition of an Integral Domain:
  • Both rings are commutative.
  • Both rings have unity.
  • Only $\mathbb{Z}$ has no zero divisors.
  • $\boxed{\therefore \mathbb{Z}\text{ is an integral domain}}$.
• $\boxed{\text{Neither ring}}$ is a field, since multiplicative inverses don’t exist for every element (since the inverse are all fractional).

Question 6.3

Given a ring $R$, for all $a,b$…

$$\begin{array}{rl}(a+b)(a-b)& =aa-ab+ba-bb\text{ by associativity}\\ & =a^2-b^2+(ba-ab)\\ & \boxed{=a^2-b^2:ba-ab⟺R\text{ is commutative }:ab=ba}\end{array}$$

Need to complete in the other direction, too?

Question 6.4

Given a ring $\mathbb{Z}/n\mathbb{Z}$, if $n$ is not prime then…

• Definition of an Integral Domain:
  • Zero divisors exist as factors of $n$, therefore $\boxed{\text{the ring is not an integral domain}}$.

Given a ring $\mathbb{Z}/p\mathbb{Z}$, if $p$ is a prime then…

• Definition of an Integral Domain:
  • The ring is commutative and has unity.
  • There are no zero divisors of $p$ as it has no factors by definition, therefore $\boxed{\text{the ring is an integral domain}}$.

Should be completed more like a mathematical proof, rather than logical.

Question 6.5

Given an integral domain $D$…

• If there exists $a\in D:a^2=1$, $a^2-1=0⟹(a+1)(a-1)=0⟺\boxed{a=\pm 1}$.
• If there exists $a\in D:a^2=a$, $a^2-a=0⟹a(a-1)=0⟺\boxed{a=0,1}$.
• If there exists $a\in D:a^n=0$ for some positive integer $n$, $a=0^{\frac{1}{n}}⟺\boxed{a=0}$.

Should state the properties of the integral domain used, e.g. commutativity when factorising.

Question 6.6

Given a ring $(R,+,\cdot ):a,b,c\in R$, then…

1. Each equation $a+x=x+a=b$ has a unique solution.
   • … proof.
2. $-(-a)=a$ and $-(a+b)=(-a)+(-b)$.
   • … proof.
3. If $m,n\in \mathbb{Z}$, then $(m+n)\cdot a=ma+na$, $m\cdot (a+b)=ma+mb$, and $m(na)=(mn)a$.
   • … proof.

Left as an exercise for exam prep.

Question 6.7

Given that the center of a ring $R$ is defined as $\left\{c\in R:cr=rc\forall r\in R\right\}$ with unity, denoted as $C$…

• Definition of a Subring (QST):
  • $C$ is non-empty, as there exists at least $r=\text{unity}$.
  • $C$ is closed under both addition and multiplication of $R$, as operating with the unity won’t change the element, hence will stay closed under $R$.
  • $C$ contains the inverse of each of its elements, by definition.

Should prove this more.

Question 6.8

The smallest subring of $\mathbb{Z}$ containing $3$ is $3\mathbb{Z}$, as it withstands QST (as it’s just a selection of the original ring, multiples of the required smallest element).

The smallest subring of $\mathbb{R}$ containing $\frac{1}{2}$ is $\mathbb{Z}[\frac{1}{2}]$, as it withstands QST (as it’s just a selection of the original ring, multiples of the required smallest element).

Should prove this more.