In a commutative ring $R$, when does the assumption $r_i\mid r$ for $1\le i\le r$ implies $\prod r_i\mid r$ (when $r_i$ are fixed)?

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

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?

In a commutative ring $R$, when the assumption $r_i\mid r$ for $1\le i\le r$ implies $\prod r_i\mid r$?

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

In a commutative ring $R$, when does the assumption $r_i\mid r$ for $1\le i\le r$ implies $\prod r_i\mid r$ (when $r_i$ are fixed)?

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