If x|y and y|x in integers, does this imply x=y?
Similarly, for any general ring, if x|y and y|x in the ring, does this imply x=y?
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Did you try $x=1,y=-1$? – David P Apr 18 '19 at 19:45
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1Consider $2$ and $-2$. If $x \mid y$ and $y \mid x$, $x$ and $y$ are called associates. – Rob Arthan Apr 18 '19 at 19:45
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Furthermore, it is equivalent to $a=ub$ for some invertible $u$ in the ring. In the case of the integers, the only invertible elements are $\pm 1$, so $a|b\iff a=\pm b$. – Yuval Gat Apr 18 '19 at 19:47
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2@YuvalGat Not generally true. It is true in domains that elements are associate iff they are unit multiples of each other, but that is not true in general rings, see here. – Bill Dubuque Apr 18 '19 at 20:05
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@BillDubuque Of course, thank you for the correction! – Yuval Gat Apr 18 '19 at 20:07