Math4302 Modern Algebra (Lecture 24)

Rings

Definition of ring

A ring is a set $R$ with binary operation $+$ and $\cdot$ such that:

• $(R,+)$ is an abelian group.
• Multiplication is associative: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
• Distribution property: $a\cdot (b+c)=a\cdot b+a\cdot c$, $(b+c)\cdot a=b\cdot a+c\cdot a$. (Note that $\cdot$ may not be abelian, may not even be a group, therefore we need to distribute on both sides.)

Properties of rings

Let $0$ denote the identity of addition of $R$. $-a$ denote the additive inverse of $a$.

• $0\cdot a=a\cdot 0=0$
• $(-a)b=a(-b)=-(ab)$, $\forall a,b\in R$
• $(-a)(-b)=ab$, $\forall a,b\in R$

Definition of commutative ring

A ring $(R,+,\cdot)$ is commutative if $a\cdot b=b\cdot a$, $\forall a,b\in R$.

Definition of unity element

A ring $R$ has unity element if there is an element $1\in R$ such that $a\cdot 1=1\cdot a=a$, $\forall a\in R$.

Definition of unit

Suppose $R$ is a ring with unity element. An element $a\in R$ is called a unit if there is $b\in R$ such that $a\cdot b=b\cdot a=1$.

In this case $b$ is called the inverse of $a$.

We use $a^{-1}$ or $\frac{1}{a}$ to represent the inverse of $a$.

Let $R$ be a ring with unity, then $0$ is not a unit. (identity of addition has no multiplicative inverse)

If $0b=b0=1$, then $\forall a\in R$, $a=1a=0a=0$.

Definition of division ring

If every $a\neq 0$ in $R$ has a multiplicative inverse (is a unit), then $R$ is called a division ring.

Definition of field

A commutative division ring is called a field.

Lemma $\mathbb{Z}_p$ is a field

$\mathbb{Z}_p$ is a field if and only if $p$ is prime.