Section 19 Integral Domains

19.2 Definition Zero Divisor

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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

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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

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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

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Link to original

19.9 Theorem

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Every field F is an integral domain

19.11 Theorem

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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 $R$

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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

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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 $\text{char}R=n$ (like $\mathbb{Z}/n\mathbb{Z}$). Proof of the second point: If $n\cdot 1=0$: $n\cdot a=a+a+\cdot \cdot \cdot +a=a(1+1+\cdot \cdot \cdot +1)=a(n\cdot 1)=a0=0$