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Let $R$ be a commutative ring with identity and $a_1, a_2, \dots,a_n \in R - \{0\}$. Then to find the gcd and lcm of $a_1, a_2, \dots,a_n$ and they exist in $R$ when $R$ is a UFD.

We can express $a_1, a_2, \dots,a_n$ uniquely as finite product of irreducible elements as:

$a_1 \sim p_1^{r_1}p_2^{r_2} \dots p_n^{r_n} $, $a_2 \sim p_1^{s_1}p_2^{s_2} \dots p_n^{s_n}$ , $\dots,$ $a_n \sim p_1^{t_1}p_2^{t_2} \dots p_n^{t_n}$

Now, we define:

$\gcd (a_1, a_2, \dots,a_n ) = p_1^{c_1}p_2^{c_2} \dots p_n^{c_n}$ where $c_i = \min\{r_i,s_i,...,t_i\}$ and,

$lcm(a_1, a_2, \dots,a_n ) = p_1^{d_1}p_2^{d_2} \dots p_n^{d_n}$ where $d_i = \max\{r_i,s_i,...,t_i\}$

I can intuitively define them but how can I prove that they exist in $R$?

Please Help!

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    $\begingroup$ Well, those formulas are in $R$ by definition, since $R$ is closed under multiplication and $p_i$ are in $R$. The real question you need to prove is, why are these the GCD and LCM. $\endgroup$ Mar 28, 2015 at 17:39
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    $\begingroup$ Incidentally, you might want to stick first with $n=2$ and then prove for general $n$ inductively, showing $\gcd(a_1,\dots a_n)=\gcd(\gcd(a_1,\cdots,a_{n-1}),a_n)$ and likewise for LCM. $\endgroup$ Mar 28, 2015 at 17:41
  • $\begingroup$ can you please help in showing that $\endgroup$
    – User8976
    Mar 28, 2015 at 18:15

1 Answer 1

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By existence and uniqueness of prime factorizations, divisibility reduces to divisibility in each prime component, i.e. $\ p^{\large a} q^{\large a'}\!\cdots\mid p^{\large b} q^{\large b'}\!\cdots\!\iff p^{\large a}\mid p^{\large b},\,\ q^{\large a'}\!\mid q^{\large b'},\,\ldots\ $

So $\,\ a\mid b,c\iff p^{\large a}\mid p^{\large b},p^{\large c},\ \ q^{\large a'}\mid q^{\large b'},q^{\large c'},\ldots\ $ each which obey

$$\quad\ \ \ \ \ \ p^{\large a}\mid p^{\large b},p^{\large c}\! \iff a\le b,c \iff a\le \min\{b,c\} \iff p^{\large a}\mid p^{\large \min\{b,c\}}$$

$\!\begin{align}{\rm So}\ \ a\mid b,c&\iff p^{\large a}\mid p^{\large \min\{b,c\}},\ q^{\large a'}\mid q^{\large \min\{b',c'\}}\ldots\\[.5em] &\iff a\mid \color{#c00}{p^{\large \min\{b,c\}} q^{\large \min\{b',c'\}}}\ldots \end{align}$

Remark $\ $ Above we employ the universal characterization / definition of gcd and lcm, i.e.

$$\begin{align} a\mid b,c\iff a\mid \color{#c00}{\gcd(b,c)}\\ b,c\mid a\iff {\rm lcm}(b,c)\mid a\end{align}\qquad\qquad$$

The $ $ lcm $ $ case is just the dual $\, \ b,c\mid a\iff \cdots,\ $ by reversing divisiblities in the gcd proof.

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