Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X37No. 420111215Ranks of modules relative to a torsion theory1933368ENSh.AsgariA.HaghanyJournal Article20090831Relative 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.http://bims.iranjournals.ir/article_368_1af8f9a81692bba1f4aa0fc6001c47b7.pdf