1. The problem statement, all variables and given/known data If A is a subring of B and B is in a particular domain like a UFD than does it imply A is also in that domain B lives in hence also a UFD? 3. The attempt at a solution Z is a ring but not a field. However Z mod p is a field. And Z mod p is a subring of Z. Although Z mod p is a ring also so a subring may live inside a more 'specialised' domain but can always be denoted the name of the domain it is a subring of. So the answer is yes to my above question.