Bézout Domains

Subjects: Ring Theory
Links: Principal Ideal Domains, Ring of Fractions, Integral Domains

Def: An integral domain $R$ in which every ideal generator by two elements is principal is called a Bézout Domain.

Obs: Every principal ideal domain is a Bézout domain.

Prop: Let $R$ be an integral domain. $R$ is a Bézout domain iff every pair of elements of $a,b$ of $R$ has a greatest common divisor $d$ in $R$ that can be written as an $R$-linear combination of $a$ and $b$, i.e., $d=ax+by$ for some $x,y\in R$.

Prop: Every finitely generated ideal of a Bézout domain is principal.

Prop: Let $F$ be the fraction field of the Bézout domain $R$. Every element of $F$ can be written in the form $a/b$ with $a,b\in R$ and $a$ and $b$ are relatively prime, i.e., $(a,b)=1$.

Th: Let $R$ be an integral domain. $R$ is a principal ideal domain iff it is a Bézout domain and unique factorisation domain.

Prop: Let $R$ be a Bézout Domain, and let $a,b,N\in R$. Consider the equation $$ax + by = N.$$ Let $d=(a,b)$ denote the greatest common divisor of $a$ and $b$ (unique up to multiplication by a unit in $R$). Then:

• Existence: The equation $ax+by=N$ has a solution $(x,y)\in {\mathbb{R}}^2$ iff $d\mid N$.
• Construction of a Particular Solution: There are $u,v\in R$ such that $$au +bv = d.$$if $d\mid N$, say $N=dm$, then $$x_0 = um,\qquad y_0 = vm $$is a particular solution to $ax+by=N$.
• General Solutions: If $(x_0,y_0)$ is one particular solution, then all solutions are given by $$ x = x_0 +\left(\frac{b}{d}\right)t, \text{ and, } y = y_0 -\left(\frac{a}{d}\right)t,$$where $t$ ranges over all elements of $R$.

The proposition above generalises the notion of a linear Diophantine equation.