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: Should I write a recommendation letter for a student at a previous job/institution?
Answer: When you change job, you don’t erase your memory (well, not in all jobs), nor do you all your responsibilities related to this previous job vanish. It is true that leaving a position creates certain obligations in the data you can retain, and how you can use non-publicly available information pertaining to your older institution. However, in the particular case of a reference letter, I don't think it should cause trouble. Moreover, reference letters are confidential.

Actually, I'd go further than saying it should not be a problem. In fact, I think if you can honestly write him a good recommendation letter, it is part of your responsibility to do it. This responsibility is not to your former institution, but to the student and the academic system as a whole.