Ranks of modules relative to a torsion theory

Document Type : Research Paper



Relative to a hereditary torsion theory $tau$ we introduce
a dimension for a module $M$, called {em $tau$-rank of} $M$,
which coincides with the reduced rank of $M$ whenever $tau$ is
the Goldie torsion theory. It is shown that the $tau$-rank of $M$
is measured by the length of certain decompositions of the
$tau$-injective hull of $M$. Moreover, some relations between the
$tau$-rank of $M$ and complements to $tau$-torsionfree
submodules of $M$ are obtained.