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I feel intuitively that for $\prod_{i\in \mathbb N}\mathbb{Z}$, as a $\mathbb{Z}$−module, and $\phi_i:\mathbb{Z}\to\mathbb{Z}$ the identity map, more than one homomorphism $\phi:\prod_{i\in \mathbb N}\mathbb{Z}\to\mathbb{Z}$ satisfies the condition "composition with the canonical embedding yields the identity map $\phi_i:\mathbb{Z}\to\mathbb{Z}$". But I can't myself define such homomorphisms $\phi:\prod_{i\in \mathbb N}\mathbb{Z}\to\mathbb{Z}$.

Could anyone suggest some appropriate functions?

Andrews
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Keith
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1 Answers1

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It is a well-known result by Specker (original reference, MO/10239) that homomorphisms $\mathbb{Z}^\mathbb{N} \to \mathbb{Z}$ are determined by what they do on the $e_i=(\delta_{ij})_j$ and that almost all $e_i$ are mapped to zero. In particular, there is no homomorphism mapping all $e_i \mapsto 1$.

Community
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Martin Brandenburg
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