MTH2001 Algebraic Structures Lecture 1

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What is a ring?

Basically a group with two operations rather than one - the more formal definition is…

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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Examples of rings?

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

What is a polynomial ring?

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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What is ring called within a ring?

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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What is an integral domain?

First, zero divisors!

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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Mathematical definition

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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This leads onto things like cancellation laws…

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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Essentially, these constructs are what we assume when doing arithmetic normally!