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Jose  
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 More options Nov 7, 10:12 pm
Newsgroups: sci.math
From: Jose <jose.fra...@gmail.com>
Date: Sat, 07 Nov 2009 12:12:19 EST
Local: Sat, Nov 7 2009 10:12 pm
Subject: Direct Sum
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Is the direct sum_{n in N} Z a countable set? i.e. infinitely many copies of Z ?

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G. A. Edgar  
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 More options Nov 8, 12:22 am
Newsgroups: sci.math
From: "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
Date: Sat, 07 Nov 2009 14:22:38 -0500
Local: Sun, Nov 8 2009 12:22 am
Subject: Re: Direct Sum
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In article
<1976345699.31113.1257613969223.JavaMail.r...@gallium.mathforum.org>,

Jose <jose.fra...@gmail.com> wrote:
> Is the direct sum_{n in N} Z a countable set? i.e. infinitely many copies of
> Z ?

What is definition of "direct sum"?
Each element has only finitely many nonzero entries?

--
G. A. Edgar                              http://www.math.ohio-state.edu/~edgar/


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Axel Vogt  
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 More options Nov 8, 12:31 am
Newsgroups: sci.math
From: Axel Vogt <&nore...@axelvogt.de>
Date: Sat, 07 Nov 2009 20:31:25 +0100
Local: Sun, Nov 8 2009 12:31 am
Subject: Re: Direct Sum
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Jose wrote:
> Is the direct sum_{n in N} Z a countable set? i.e. infinitely many copies of Z ?

You may mean either the one or the other:

Z^(IN) = sum or
Z^IN   = product (with infinitely many copies)


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Arturo Magidin  
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 More options Nov 8, 1:07 am
Newsgroups: sci.math
From: Arturo Magidin <magi...@member.ams.org>
Date: Sat, 7 Nov 2009 12:07:13 -0800 (PST)
Local: Sun, Nov 8 2009 1:07 am
Subject: Re: Direct Sum
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On Nov 7, 11:12 am, Jose <jose.fra...@gmail.com> wrote:

> Is the direct sum_{n in N} Z a countable set? i.e. infinitely many copies of Z ?

If by "direct sum" you mean what is usually meant, to wit, the
subgroup of the direct product of elements of finite support (almost
all entries are equal to 0), then the answer is yes. In fact, it is a
free abelian group of rank aleph_0, and therefore has aleph_0*aleph_0
= (aleph_0)^2 = aleph_0 elements.

If you mean the unrestricted direct product, then no; the direct
product of countably many copies of Z has aleph_0^{aleph_0) = 2^
{aleph_0) elements.

--
Arturo Magidin


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