Question

Let $\mathit{R}$ be a PID and $\mathit{S}$ an integral domain that contains $\mathit{R}$. Let $\mathit{a}\mathbf{,}\mathit{b}\mathbf{,}\mathit{d}\mathbf{\in }\mathbf{ℝ}$. If d is a gcd of a and b in $\mathit{R}$, prove that d is a gcd of a and b in $\mathit{S}$ .

Short answer

If d is a gcd of a and b in$R$then d is a gcd of a and b in $S$.

Step-by-step solution

Exercise 20

If $R$ be a PID and$a,b\in R$not both zero then a, b have a greatest common divisor that can be written as a linear combination of a and b.

gcd of a and b in R

Let $R$ be a PID and d is a gcd of $a,b\in R$. Thus, d divides both a and b.

Since, $R$ is contained in $S$it divides both a and b in S.

From exercise 20, $R$ is a PID then there exists$p,q\in R$ such that $d=ap+bq$…… (1)

gcd of a and b in S

Let c be a common divisor of a and b in $S$ then there exists m, n such that $a=cm$and$b=cn$.

Substitute the values of a and b in (1).

$\begin{array}{c}d=ap+bq\\ =cmp+cnq\\ =c(mp+nq)\end{array}$

Thus, c divides d in S then by definition d is the gcd of a, b in$S$.

.

Therefore,if d is a gcd of a and b in$R$then d is a gcd of a and b in $S$.