Is the zero module semisimple?
- From: "Anvita" <anvita21@xxxxxxx>
- Date: 8 Oct 2006 20:33:54 -0700
The zero module is not considered simple. However, no textbook
available to me says explicitly whether it is semisimple.
My gut reaction would be not to label it semisimple,
but on second thought...
There are several (equivalent) definitions of a semisimple module.
Two of them are as follows:
(1) A module M is semisimple iff, for every submodule N, there is
a submodule K such that M is the direct sum of N and K.
(2) A module M is semisimple iff it is the direct sum of simple
modules.
The zero module qualifies for (1), but I am having trouble seeing
whether it does for (2). It boils down to deciding what the direct sum
of the empty set of modules is. Something tells me that it should be
the zero module. Can this be proven?
Thank you
Anvita
.
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