What are all the homomorphisms between the rings Z18 and Z15?Any homomorphism ψ between the rings Z18 and Z15 is completely defined by ψ(1). So from0=φ(0)=φ(18)=φ(18⋅1)=18⋅φ(1)=15⋅φ(1)+3⋅φ(1)=3⋅φ(1)We get that ψ(1) is either 5 or 10. But how can I prove or disprove that these two are valid homomorphisms?

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What are all the homomorphisms between the rings Z18 and Z15?Any homomorphism ψ between the rings Z18 and Z15 is completely defined by ψ(1). So from0=φ(0)=φ(18)=φ(18⋅1)=18⋅φ(1)=15⋅φ(1)+3⋅φ(1)=3⋅φ(1)We get that ψ(1) is either 5 or 10. But how can I prove or disprove that these two are valid homomorphisms?

entreblogsmc2j · Accepted Answer

If one has a homomorphism of two rings R,S, and R has an identity, then the identity must be mapped to an idempotent element of S, because the equation&amp;nbsp;x2=x&amp;nbsp;is preserved under homomorphisms. Now 5 is not an idempotent element in&amp;nbsp;ℤ15, so the map generated by&amp;nbsp;1→5&amp;nbsp;is not a homomorphism.However, 10 is an idempotent element of&amp;nbsp;ℤ15. In particular, the subring&amp;nbsp;T⊂ℤ15&amp;nbsp;generated by 10 has unit 10. Since it is annihilated by 3, and consequently by 18, there is a unital homomorphism&amp;nbsp;ℤ18→T&amp;nbsp;(i.e., mapping 1 to 10). So your second map is a legitimate homomorphism of rings (composing with the injection&amp;nbsp;T→ℤ15).Basically, the point of this answer is to check that one of your maps preserves the relations of the two rings, while the other doesn&#039;t.

What are all the homomorphisms between the rings \mathbb{Z}_{18}\ \text{and}\ \mathbb{Z}_{15}?

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