TITLE: Is it possible that $\text{lcm}(zx, zy)$ exists but $\text{lcm}(x, y)$ not?
QUESTION [6 upvotes]: Is it possible to find an example of an integral domain $D$ and a pair of non-zero elements $x$ and $y$ in $D$ such that $\text{lcm}(zx, zy)$ exists for some non-zero element $z$ in $D$ but $\text{lcm}(x, y)$ does not?
Definition of least common divisor (lcm):
Let $a$ and $b$ be elements of a commutative ring $R$. A common multiple of $a$ and $b$ is an element $m$ of $R$ such that there exist elements $x$ and $y$ of $R$ such that $ax = by = m$. A least common multiple of $a$ and $b$ is a common multiple $m$ of $a$ and $b$ that is minimal in the sense that for any other common multiple $n$ of $a$ and $b$, $n=zm$ for some $z$ in $R$.

REPLY [5 votes]: Let $M$ be a lowest common multiple of $zx$ and $zy$. It's clear that $z$ divides $M$, say $M=zm$, so we can try and see whether $m$ satisfies the properties for being a lowest common multiple of $x$ and $y$.
First, $M=azx=bzy$, so $m=ax=by$.
Now suppose $r$ is a common multiple of $x$ and $y$. Then $zr$ is a common multiple of $zx$ and $zy$; therefore $M=zm$ divides $zr$. Hence $m$ divides $r$.