A non-zero module which is not the direct sum of two of its proper submodules. The negation of indecomposable is, of course, decomposable. An abstract vector space is indecomposable iff it has dimension 1.
As a consequence of Kronecker basis theorem, an Abelian group is indecomposable iff
it is either isomorphic to 
or to 
, where 
is a prime power. This is not
the case for 
,
and in fact we have
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