In what kind of rings a divisor of a product is a product of divisors? In a unique factorisation domain, if $a|bc$, then $a$ can be written as $a = a_1a_2$ so that $a_1|b$ and $a_2|c$.
Is this property of a commutative ring strictly weaker than the property of being a UFD?  If so, does it have a name, or is it equivalent to some other well-known property?
 A: Such rings are known as pre-Schreier rings. The property easily yields that atoms (irreducibles) are prime, so it is equivalent to being a UFD in atomic domains (nonzero nonunits factor into atoms), but is weaker in absence of that. Below is a summary of related properties of domains.
PID:  $\ \ $  every ideal is principal 
Bezout: $\ \ $ every ideal (a,b) is principal 
GCD: $\ \ $   (x,y) := gcd(x,y) exists for all x,y 
SCH:  $\ \ $  Schreier = pre-Schreier & integrally closed 
SCH0: $\ \ $  pre-Schreier: a|bc  $\, \Rightarrow\, $  a = BC, B|b, C|c 
D:  $\ \ $    (a,b) = 1  &  a|bc  $\,\Rightarrow\,$  a|c 
PP:  $\ \ $   (a,b) = (a,c) = 1   $\,\Rightarrow\,$  (a,bc) = 1 
GL:  $\ \ $   Gauss Lemma: product of primitive polys is primitive 
GL2:  $\ \ $  Gauss Lemma holds for all polys of degree 1 
AP: $\ \ $    atoms are prime [AP = PP restricted to atomic a] 

Since  atomic & AP $\,\Rightarrow\,$ UFD, reversing the above  UFD $\,\Rightarrow\,$ AP  path shows that in atomic domains all these properties (except PID, Bezout) collapse, becoming all equivalent to UFD. 
There are also many properties known equivalent to D, e.g. 
[a] $\ \ $    (a,b) = 1  $\,\Rightarrow\,$     a|bc  $\,\Rightarrow\,$   a|c 
[b] $\ \ $    (a,b) = 1  $\,\Rightarrow\,$    a,b|c  $\,\Rightarrow\,$  ab|c 
[c] $\ \ $    (a,b) = 1  $\,\Rightarrow\,$  (a)/\(b) =  (ab) 
[d] $\ \ $    (a,b) exists  $\,\Rightarrow\,$  lcm(a,b) exists 
[e]  $\ \ $   a + b X  irreducible  $\,\Rightarrow\,$  prime  for b $\ne$ 0 (deg = 1) 
You can find proofs of most of the above (including counterexamples for implication reversals) by looking up papers by D. Anderson and M. Zafrullah with the keywords "Schreier" and "Gauss's Lemma". See also this answer for links on related topics such as the classical Euler four number theorem (Vierzahlensatz), Riesz interpolation, etc.
