In a commutative ring $R$, when does the assumption $r\_i\mid r$ for $1\le i\le n$ imply $\prod\_{1\le i\le n} r\_i\mid r$ (when $r\_i$ are fixed)?


Does there exist any criterion for this implication that is related to regular sequences?



When an element of a ring that is divisible by a finite set of elements is necessarily divisible by their product?
This is just an expansion of my comment. Question is local, so we may assume that the ring is local. If $r\_1,\ldots, r\_n$ is a regular sequence, then they are so in any order and any subset forms a regular sequence. So, assume $r\_i|r$ for all $i$ and assume we have shown $r=ar\_1r\_2\cdots r\_i$ for some $i<n$, the case of $i=1$ being given.


Then $r\_1,\ldots, r\_i$ is a regular sequence modulo $r\_{i+1}$. Then $r\_1\cdots r\_i$ is also a regular sequence modulo $r\_{i+1}$ and then since $r\_{i+1}|r$ implies $r\_{i+1}|a$ and by induction we are done.


