# Showing that the direct product does not satisfy the universal property of the direct sum

I feel intuitively that for $\prod_{i\in N}\mathbb{Z}$, as a $\mathbb{Z}$−module, and $\phi_i:\mathbb{Z}\to\mathbb{Z}$ the identity map, more than one homomorphism $\phi:\prod_{i\in N}\mathbb{Z}\to\mathbb{Z}$ 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 $\phi:\prod_{i\in N}\mathbb{Z}\to\mathbb{Z}$.

Could anyone suggest some appropriate functions?

#### Solutions Collecting From Web of "Showing that the direct product does not satisfy the universal property of the direct sum"

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 $e_i \mapsto 1$.