1. The problem statement, all variables and given/known data Can there be a mapping that may not map any elements from one domain to another? The reason is that the mapping has a condition. For example, it will only map elements if the one in the domain are related in some way to the element they are mapped to (i.e congruence via a certain ideal). If this specified relation dosen't hold then no mapping will occur. If the relation is specified than the mapping certainly obeys a homomorphism. For a concrete example, say you map Z[x] to Z via the identity transformation. Then clearly polynomials of degree 1 or more in Z[x] will not be mapped to Z. Are they just left alone? Is this transformation still valid?