# Construct a homomorphism

I try to construct a homomorphism $$\mathbb{Z}[x] \rightarrow \mathbb{Z}[D_p]$$ which is surjective and where $D_p$ is the dihedral group, $\mathbb{Z}[D_p]$ a monoidring . My problem is that $D_p$ has two generators. For groups with one generator I can make it.

Thank you very much

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Maybe I better add the motivation for the question: I am looking for isomorphisms like $\mathbb{Z}[x](X^p-1) \cong \mathbb{Z}[C_p]$ for other than cyclic groups. – user56913 Jan 11 '13 at 12:01
What kind of homomorphism are you looking for ? Note that there couldn't be a surjective ring homomorphism, since $\mathbb{Z}[x]$ is commutative, but $\mathbb{Z}[D_p]$ isn't. – Ralph Jan 11 '13 at 12:20
... But $\mathbb{Z}[x]$ and $\mathbb{Z}[D_p]$ are isomorphic as additive abelian groups (since as such either is free abelian of countably infinite rank). – Ralph Jan 11 '13 at 12:24
Maybe I am misunderstanding you question but there can be no surjective morphism from a monoid with one generator a monoid that requires two generators since this would give you a surjective moprhism from a free monoid on one generator to free monoid on two generators. It should be obvious that such a morphism doesn't exist. Therefore, in general, $\mathbb Z [x]$ can surjectively map only into $\mathbb Z[C_p]$, right? – Marek Jan 11 '13 at 12:27
I would have needed a ringhomomorphism. So there is also no isomorphism for $D_p$ like in my first comment? – user56913 Jan 11 '13 at 12:30