MTH2001 Algebraic Structures Pre-Lecture 1

• Lecture Notes.
• Lecture Notes (Detailed).
• Lecture: MTH2001 Algebraic Structures Lecture 1.
• Practical (MTH2001 Algebraic Structures Practical 1).

KEY DIFFERENCE BETWEEN GROUPS AND RINGS: Rings have two operations, rather than one!

Integral domains then just extend rings with unity and commutativity!

Definition of a Ring

A ring is a set $R$ with two operations, usually denoted by $+$ and $\cdot$, satisfying:

• $R$ with $+$ is an abelian group.
• $R$ is closed with respect $\cdot$, i.e. $a\cdot b\in R,\forall a,b\in R$.
• $\cdot$ is associative: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
• For all $a,b,c\in R$, the distributive laws hold:
  • $a\cdot (b+c)=a\cdot b+a\cdot c$.
  • $(a+b)\cdot c=a\cdot c+b\cdot c$.

Notation: Suppose a set $R$. If $R$ is a ring with operations $⨁$ and $⨀$, we write $(R,⨁,⨀)$. If the multiplicative operator is omitted, the group $(R,+)$ is an additive group.

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There are many examples of rings, such as $(\mathbb{Z},+,\cdot )$, $(\mathbb{Q},+,\cdot )$, $(\mathbb{R},+,\cdot )$, $(\mathbb{C},+,\cdot )$, and even polynomial rings:

Definition of a Polynomial Ring

Let $\mathbb{R}[x]$ be the set of all polynomials in $x$:

$$\mathbb{R}[x]=\left\{a_0,a_{1}x,\dots ,a_n{x}^n:a_n\in \mathbb{R},n\in \mathbb{N}\cup \left\{0\right\}\right\}$$

This is a ring with sum and multiplication of polynomials:

$$(f+g)(x)=f(x)+g(x)\text{and}(f\cdot g)(x)=f(x)\cdot g(x)$$

Note: all polynomial rings are commutative rings, i.e. $a\cdot b=b\cdot a$.

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There are also rings within rings, called:

Definition of a Subring

Let $(R,+,\cdot )$ be a ring. We say that a subset $S\subset R$ is a subring if $(S,+,\cdot )$ is itself a ring.

Notation: $S\le R$.

Quick way to test a subset:

Definition of Quick Subring Theorem

A subset $S$ of a ring $R$ is a subring if and only if…

1. $S$ is non-empty.
2. $S$ is closed under both addition and multiplication of $R$.
3. $S$ contains the negative (i.e. the additive inverse) of each of its elements.

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The latter mentioned QST can be shown to prove that the Gaussian numbers $\mathbb{Z}[i]=\left\{a+bi:a,b\in \mathbb{Z}\right\}$ form a subring of $\mathbb{C}$, for example. This can be done directly (substitute expressions into definitions and draw logical conclusions).

An additional construct is the concept of zero divisors. For example, in integers if you multiply non-zero numbers, you’ll obtain a non-zero numbers.

Under modulo though, there are non-zero numbers that multiply to give zero: these are zero divisors. E.g.,

$$\bar{2}\ne \bar{0}\text{but}\bar{2}\cdot \bar{2}=\bar{4}=\bar{0}$$

Definition of a Zero Divisor

Let $R$ be a commutative ring. We say that an element $r\in R$ is a zero divisor if $a\cdot b=0$ for some element $b\ne 0$ of $R$.

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Finally…

Definition of an Integral Domain

If we have a commutative ring with unity $R$, then $R$ is an integral domain if it has no zero divisors (e.g. $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$):

• Commutative,
• Unity is $1$,
• $a\cdot b=0⟺a=0\text{ or }b=0$.

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Note: a unity is basically an identity element and zero element (additive and multiplicative unities respectively).

Lots of commutative modulo rings aren’t integral domains, and naturally lots of non-commutative rings aren’t either.

Exercise: Show that if $n$ is not a prime, then $\mathbb{Z}/n\mathbb{Z}$ is not an integral domain.

Exercise: Show that if $p$ is prime, then $\mathbb{Z}/p\mathbb{Z}$ is an integral domain.

In these integral domains, there are cancellation laws that work.

Definition of Cancellation Laws

Let $D$ be an integral domain with $a,b,c\in D$. If $a\ne 0$, then $ab=ac⟹b=c$. Similarly, $ba=ca⟹b=c$.

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This is quite trivial to prove, also, as it’s just a natural assumption taught in elementary algebra. However, if $R$ is a ring that is not an integral domain, the laws might not work! For example, under $\mathbb{Z}/12\mathbb{Z}$.