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












4












$begingroup$


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



Could anyone suggest some appropriate functions?






abstract-algebra category-theory modules







share|cite|improve this question











$endgroup$












  • $begingroup$
    By $N$ you mean the set of natural numbers?
    $endgroup$
    – Daniel Fischer♦
    Sep 26 '14 at 15:13
















4












$begingroup$


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



Could anyone suggest some appropriate functions?






abstract-algebra category-theory modules







share|cite|improve this question











$endgroup$












  • $begingroup$
    By $N$ you mean the set of natural numbers?
    $endgroup$
    – Daniel Fischer♦
    Sep 26 '14 at 15:13














4












4








4


0



$begingroup$


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



Could anyone suggest some appropriate functions?






abstract-algebra category-theory modules







share|cite|improve this question











$endgroup$




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



Could anyone suggest some appropriate functions?






abstract-algebra category-theory modules




abstract-algebra category-theory modules






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 17 '18 at 7:20









Andrews

1,2691421




1,2691421










asked Sep 26 '14 at 14:44









KeithKeith

1,4251122




1,4251122












  • $begingroup$
    By $N$ you mean the set of natural numbers?
    $endgroup$
    – Daniel Fischer♦
    Sep 26 '14 at 15:13


















  • $begingroup$
    By $N$ you mean the set of natural numbers?
    $endgroup$
    – Daniel Fischer♦
    Sep 26 '14 at 15:13
















$begingroup$
By $N$ you mean the set of natural numbers?
$endgroup$
– Daniel Fischer♦
Sep 26 '14 at 15:13




$begingroup$
By $N$ you mean the set of natural numbers?
$endgroup$
– Daniel Fischer♦
Sep 26 '14 at 15:13










1 Answer
1






active

oldest

votes


















1












$begingroup$

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 $e_i mapsto 1$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    where can I find the proof about there is no homomorphism mapping all e i ↦1? The original reference seems to be written in German..Could you suggest me some English links?
    $endgroup$
    – Keith
    Sep 26 '14 at 17:12










  • $begingroup$
    I have included a link to MO.
    $endgroup$
    – Martin Brandenburg
    Sep 26 '14 at 17:13










  • $begingroup$
    Oh didn't recognize it. Thank you :)
    $endgroup$
    – Keith
    Sep 26 '14 at 17:17











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1 Answer
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

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 $e_i mapsto 1$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    where can I find the proof about there is no homomorphism mapping all e i ↦1? The original reference seems to be written in German..Could you suggest me some English links?
    $endgroup$
    – Keith
    Sep 26 '14 at 17:12










  • $begingroup$
    I have included a link to MO.
    $endgroup$
    – Martin Brandenburg
    Sep 26 '14 at 17:13










  • $begingroup$
    Oh didn't recognize it. Thank you :)
    $endgroup$
    – Keith
    Sep 26 '14 at 17:17
















1












$begingroup$

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 $e_i mapsto 1$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    where can I find the proof about there is no homomorphism mapping all e i ↦1? The original reference seems to be written in German..Could you suggest me some English links?
    $endgroup$
    – Keith
    Sep 26 '14 at 17:12










  • $begingroup$
    I have included a link to MO.
    $endgroup$
    – Martin Brandenburg
    Sep 26 '14 at 17:13










  • $begingroup$
    Oh didn't recognize it. Thank you :)
    $endgroup$
    – Keith
    Sep 26 '14 at 17:17














1












1








1





$begingroup$

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 $e_i mapsto 1$.






share|cite|improve this answer











$endgroup$



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 $e_i mapsto 1$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Apr 13 '17 at 12:58









Community♦

1




1










answered Sep 26 '14 at 15:17









Martin BrandenburgMartin Brandenburg

108k13165335




108k13165335












  • $begingroup$
    where can I find the proof about there is no homomorphism mapping all e i ↦1? The original reference seems to be written in German..Could you suggest me some English links?
    $endgroup$
    – Keith
    Sep 26 '14 at 17:12










  • $begingroup$
    I have included a link to MO.
    $endgroup$
    – Martin Brandenburg
    Sep 26 '14 at 17:13










  • $begingroup$
    Oh didn't recognize it. Thank you :)
    $endgroup$
    – Keith
    Sep 26 '14 at 17:17


















  • $begingroup$
    where can I find the proof about there is no homomorphism mapping all e i ↦1? The original reference seems to be written in German..Could you suggest me some English links?
    $endgroup$
    – Keith
    Sep 26 '14 at 17:12










  • $begingroup$
    I have included a link to MO.
    $endgroup$
    – Martin Brandenburg
    Sep 26 '14 at 17:13










  • $begingroup$
    Oh didn't recognize it. Thank you :)
    $endgroup$
    – Keith
    Sep 26 '14 at 17:17
















$begingroup$
where can I find the proof about there is no homomorphism mapping all e i ↦1? The original reference seems to be written in German..Could you suggest me some English links?
$endgroup$
– Keith
Sep 26 '14 at 17:12




$begingroup$
where can I find the proof about there is no homomorphism mapping all e i ↦1? The original reference seems to be written in German..Could you suggest me some English links?
$endgroup$
– Keith
Sep 26 '14 at 17:12












$begingroup$
I have included a link to MO.
$endgroup$
– Martin Brandenburg
Sep 26 '14 at 17:13




$begingroup$
I have included a link to MO.
$endgroup$
– Martin Brandenburg
Sep 26 '14 at 17:13












$begingroup$
Oh didn't recognize it. Thank you :)
$endgroup$
– Keith
Sep 26 '14 at 17:17




$begingroup$
Oh didn't recognize it. Thank you :)
$endgroup$
– Keith
Sep 26 '14 at 17:17


















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