Abstract algebra/Integral domains

Introduction to Integral Domains

Motivation: The concept of divisibility is central to the study of ring theory. Integral domains are a useful tool for studying the conditions under which concepts like divisibility and unique factorization are well-behaved.

Definition An integral domain is a commutative ring $R$ with $1_R\ne 0_R$ such that for all $a,b\in R$, the statement $ab=0$ implies either $a=0$ or $b=0$.

An equivalent definition is as follows:

Definition Given a ring $R$, a zero-divisor is an element $a\in R$ such that $\mathrm{\exists }x\in R,x\ne 0$ such that $a*x=0_R$.

Definition An integral domain is a commutative ring $R$ with $1_R\ne 0_R$ and with no non-zero zero-divisors.

Remark An integral domain has a useful cancellation property: Let $R$ be an integral domain and let $a,b,c\in R$ with $a\ne 0$. Then $ab=ac$ implies $b=c$. For this reason an integral domain is sometimes called a cancellation ring.

Examples:

1. The set $\mathbb{Z}$ of integers under addition and multiplication is an integral domain. However, it is not a field since the element $2\in Z$ has no multiplicative inverse.
2. The set trivial ring {0} is not an integral domain since it does not satisfy $0\ne 1$.
3. The set ${\mathbb{Z}}_6$ of congruence classes of the integers modulo 6 is not an integral domain because $[2]*[3]=[0]$ in ${\mathbb{Z}}_6$.

Theorem: Any field $F$ is an integral domain.

Proof: Suppose that $F$ is a field and let $a\in F,a\ne 0$. If $ax=0$ for some $x$ in $F$, then multiply by $a^{-1}$ to see that $ax=0\Rightarrow a^{-1}(ax)=a^{-1}0\Rightarrow 1x=0\Rightarrow x=0$. $F$ cannot, therefore, contain any zero divisors. Thus, $F$ is an integral domain.$◻$

Definition If $R$ is a ring, then the set of polynomials in powers of $x$ with coefficients from $R$ is also a ring, called the polynomial ring of $R$ and written $R[x]$. Each such polynomial is a finite sum of terms, each term being of the form $r{x}^n$ where $r\in R$ and $x^n$ represents the $n$-th power of $x$. The leading term of a polynomial is defined as that term of the polynomial which contains the highest power of $x$ in the polynomial.

Remark A polynomial equals $0$ if and only if each of its coefficients equals $0$.

Theorem: Let $R$ be an integral domain and let $R[x]$ be the ring of polynomials in powers of $x$ whose coefficients are elements of $R$. Then $R[x]$ is an integral domain if and only if $R$ is.

Proof If commutative ring $R$ is not an integral domain, it contains two non-zero elements $a$ and $b$ such that $ab=0$. Then the polynomials $ax$ and $bx$ are non-zero elements of $R[x]$ and $axbx=abxx=0xx=0$. Thus if $R$ is not an integral domain, neither is $R[x]$.

Now let $R$ be an integral domain and let $A$ and $B$ be polynomials in $R[x]$. If the polynomials are both non-zero, then each one has a non-zero leading term, call them $a{x}^m$ and $b{x}^n$. That these are the leading terms of polynomials $A$ and $B$ means that the leading term of the product $AB$ of these polynomials is $ab{x}^{m+n}$. Since $R$ is an integral domain and $a,b\in R$, $ab\ne 0$. This means, by the Remark above, that the product $AB$ is not zero either. This means that $R[x]$ is an integral domain.