GCD of multiple elements in a UFD

Question

Suppose that the GCD of $a_1,..,a_r$ is a unit. Show that the GCD of $ka_1,..,ka_r$ is $k$, where $k\in R$.

Its seems simple enough but I can't seem to show that any divisor of $ka_1,..,ka_r$ can divide $k$, i.e. i tried to show the (k) was the smallest principal ideal that contains $ka_1,..,ka_r$.

Can you just show for the case $r = 2$ and then use induction on $r$? — Azlif, Jan 08 '18 at 14:21

score 0 · Accepted Answer · answered Jan 08 '18 at 14:29

0

The $r=2$ case follows from a theorem dealt with here; induction (or extending the methodology there) will help with the general case.

answered Jan 08 '18 at 14:29

BallBoy

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GCD of multiple elements in a UFD

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