From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: direct product of semisimple module
Date: Sat, 22 Jul 2000 06:56:09 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8lao6r$11a61@hkunae.hku.hk>,
  Siu Lok Shun <lssiu@hkusua.hku.hk> wrote:
> It is a standard result that every direct sum of semisimple modules is
> semisimple. I guess this is not the case for a direct product of
> infinitely many summands. But I cannot think of a counter example, who can
> help me?

Let your ring be Z and consider the diresct product of Z/pZ over
all primes p.

--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
 "`The twenty-first century didn't begin until a minute
  past midnight January first 2001.'"
   John Brunner, _Stand on Zanzibar_ (1968)


Sent via Deja.com http://www.deja.com/
Before you buy.
==============================================================================

From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: direct product of semisimple module
Date: Sat, 22 Jul 2000 17:37:27 GMT
Newsgroups: sci.math

In article <157563f0.03f595b6@usw-ex0108-061.remarq.com>,
  Alex K <akocsardNOakSPAM@fceia.unr.edu.ar.invalid> wrote:
> In Hungerford (Algebra), you can find that the radical of a
> product of rings is the product of the radicals. So the
> product of a family of semisimple rings is semisimple.

For a start, the original question was about semisimple *modules*,
not semisimple rings. Also it is false that a product of
semisimple rings has to be semisimple. Note that a ring is semisimple
iff it has zero radical *and* it is Artinian. The Artinian condition
is not preserved by infinite products. Indeed it is obvious
that an infinite product of nonzero rings cannot be Artinian and
so cannot be semisimple either!

--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
 "`The twenty-first century didn't begin until a minute
  past midnight January first 2001.'"
   John Brunner, _Stand on Zanzibar_ (1968)


Sent via Deja.com http://www.deja.com/
Before you buy.
==============================================================================

From: Siu Lok Shun <lssiu@hkusua.hku.hk>
Subject: Re: direct product of semisimple module
Date: 23 Jul 2000 01:07:04 GMT
Newsgroups: sci.math

Robin Chapman <rjc@maths.ex.ac.uk> wrote:
> In article <157563f0.03f595b6@usw-ex0108-061.remarq.com>,
>   Alex K <akocsardNOakSPAM@fceia.unr.edu.ar.invalid> wrote:
> > In Hungerford (Algebra), you can find that the radical of a
> > product of rings is the product of the radicals. So the
> > product of a family of semisimple rings is semisimple.

> For a start, the original question was about semisimple *modules*,
> not semisimple rings. Also it is false that a product of
> semisimple rings has to be semisimple. Note that a ring is semisimple
> iff it has zero radical *and* it is Artinian. The Artinian condition
> is not preserved by infinite products. Indeed it is obvious
> that an infinite product of nonzero rings cannot be Artinian and
> so cannot be semisimple either!

Even the infinitely sum of semisimple ring is not semisimple because of
possessing infinitely many nonisomorphic minimal ideals.

However I had thought of your suggested example \Prod ^infty Z_p As a Z
module. I find difficult to think of the general form of the minimal
Z-module or the maximal Z-module.

> --
> Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
>  "`The twenty-first century didn't begin until a minute
>   past midnight January first 2001.'"
>    John Brunner, _Stand on Zanzibar_ (1968)


> Sent via Deja.com http://www.deja.com/
> Before you buy.

--