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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?

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# Homework Help: Valid mapping?

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