Section 19 Integral Domains

19.2 Definition Zero Divisor

[[Math/ Books/A First Course in Abstract Algebra.pdf#page=183&selection=250,1,285,1&color=note|p.178]]

If a and b are two nonzero elements of a ring R such that ab = 0, then a and b are divisors of 0 (or 0 divisors).

19.3 Theorem

[[Math/ Books/A First Course in Abstract Algebra.pdf#page=183&selection=304,1,316,1&color=note|p.178]]

In the ring ${\mathbb{Z}}_n$ , the divisors of 0 are precisely those nonzero elements that are not relatively prime to n.

Let $m\in {\mathbb{Z}}_n$, $m\ne 0$ and $d=gcd(n,m)>1$ $m(\frac{n}{d})=n(\frac{m}{d})$ $n(\frac{m}{d})$ is 0 because $0\equiv n(\frac{m}{d})modn$ , so m is a divisor of 0.

If $gcd(n,m)=1$ then $m\frac{n}{1}=mn=m0=0$

19.5 Definition Integral Domain

p.179

An integral domain D is a commutative ring with unity $1\ne 0$ and containing no divisors of 0.

This means we can factorize a polynomial and solve an equation as we do normally. There is no other other solutions than $x=r_1$ and $x=r_2$ $(x-r_1)(x-r_2)=0$

NOTE

${\mathbb{Z}}_n$ is an integral domain if and only if $n$ is prime

NOTE

A direct product of two nonzero rings $R$ and $S$ cannot be an integral domain, because $(r,0)(s,0)=(0,0)$ for $r\in R$ and $s\in S$

Algebra Structure.excalidraw

⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠ You can decompress Drawing data with the command palette: ‘Decompress current Excalidraw file’. For more info check in plugin settings under ‘Saving’

Excalidraw Data

Text Elements

Monoid

Semigroup

Group

Field

• identity element

• inverses

Abelian Group

• commutativity

• second operation (multiplication)

• commutative multiplication

Division Ring (Skew Field)

Commutative Ring

• commutative multiplication

• invertibility (non-zero)

Ring

• multiplicative identity

Magma

Set

• binary operation

• associativity

Rng (without Unity)

Semiring

• additive inverses

• associative& distributive

• second operation (multiplication)

• associative& distributive

• invertibility (non-zero)

Integral Domain

• no zero-divisors

• invertibility (non-zero)

Commutative Monoid

• commutativity

• the absorption property of zero

Link to original

19.9 Theorem

p.180

Every field F is an integral domain

19.11 Theorem

p.181

Every finite integral domain is a field.

• If p is a prime, ${\mathbb{Z}}_p$ is a field

19.13 Definition The Characteristic of the ring $R$

p.181

If for a ring R a positive integer n exists such that n · a = 0 for all a ∈ R, then the least such positive integer is the characteristic of the ring R. If no such positive integer exists, then R is of characteristic 0.

$\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$ all have characteristic 0 ${\mathbb{Z}}_n$ is of characteristic n

19.15 Theorem

p.182

Let R be a ring with unity.

• If $n\cdot 1\ne 0$ for all $n\in {\mathbb{Z}}^{+}$, then R has characteristic of 0.
• If $n\cdot 1=0$ for some $n\in {\mathbb{Z}}^{+}$, then the smallest such integer n is the characteristic of R. You can think of the characteristic of a ring $R$ very simply as “how many times you must add $1$ to itself before you get $0$”:

• Keep adding $1$: 1,  1+1,  1+1+1,  …
• If you never hit $0$, the ring has characteristic $0$ (like $\mathbb{Z}$).
• If the first time you reach $0$ is after $n$ summands, and no smaller positive sum vanishes, then $\mathrm{char}R=n$ (like $\mathbb{Z}/n\mathbb{Z}$). Proof of the second point: If $n\cdot 1=0$: $n\cdot a=a+a+\cdot \cdot \cdot +a=a(1+1+\cdot \cdot \cdot +1)=a(n\cdot 1)=a0=0.$