Ranks of modules relative to a torsion theory

Abstract

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