i understand that if two rings are isomorphic, and one ring is an integral domain, so must the other be.
however, consider two rings, both commutative rings with unity.  is it possible that one ring contains zero divisors and one does not while there exists an ring homomorphism between the two?
there could not be a isomorphism between the two rings because there would be no one to one or onto mapping between the two rings. but could there be an operation preserving mapping between an integral domain and a commutative ring with unity and with zero divisors? 
clearly if such a mapping did exists it would not seem to be one to one or onto, but does this rule out the potential of a homomorphism existing between the two?


Question: Is there words that mean "meet the criteria"?
Answer: You could use "comply", which means "meet specified standards."

For a similar word in theme of hair, how about adhere? 

Adhere or rather "adhere to" is a synonym of comply.